The Greatest Women Mathematicians

I have to admit a reluctance to putting the modifier “Women” in this post, because it would seem to imply that on an absolute scale the mathematicians I mention here are not intrinsically great. Perhaps a better title would be The Greatest Mathematicians (who happen to be Women). In any case, it has always bothered me when I see girls and women either discouraged from or outright forbidden from becoming mathematicians. One way or another, it is I think a sign of our times that “mathematicians you’ve never heard of” is kind of redundant.

I present these mathematicians in no particular order, for many reasons. Among those reasons is my personal opinion that the field of mathematics itself is not (again popular notions notwithstanding) like a vertical ladder, where first you learn counting, arithmetic, then algebra, geometry, trig, calculus and so on. In fact Mathematics, like Art, is more of a tree, with many branches, and many ways of thinking and seeing things. Some math is visual, some verbal, even some tactile. The fields these women pursued were likewise in many different areas, and their peculiar genius or accomplishment in each was profound. I won’t talk about all of the women pictured above, just the ones about which I would like to make a point. If you like, google “Greatest Women Mathematicians” for a very long and interesting list.

Maryam Mirzakhani

When I was writing this piece this morning I was shocked and saddened to see that Maryam Mirzakhani had just died last year (2017) of breast cancer. She was only 40, but had already done some profound work in geometry, especially Riemannian geometry — used by physicists in general relativity and elsewhere. She won the Fields Medal for her work in 2014, and became the first woman in history to win this award, described as the “Nobel Prize in Mathematics”. Maryam was born in Iran, and upon news of her death, a number of Iranian newspapers broke the taboo of printing a picture of her (a woman) with her hair uncovered.

Cathleen Synge Morawetz

Just one month after Maryam Mirzakhani died, we also lost Cathleen Morawetz (1923-2017), Professor Emeriti at New York University. Unlike most of the other mathematicians in this list, I had the great fortune to meet and get to know Cathleen in the 1980’s, while doing postdoc work at the Courant Institute in New York, where she at the time was the Director.

I had gone to Courant to continue my studies of nonlinear wave equations, and Cathleen had made much of her own fame in that area, studying compressible fluids and shock waves. She was also the creator of the “Morawetz Inequality(ies)”, which have proven to have many uses, even to the understanding the stability of Black Holes.

Cathleen was a very smart and jovial woman, and I will miss her.

Emmy Noether

Going back a bit, it would be difficult to convey just how profound and far-reaching was the work done by Emmy Noether, who lived from 1882 to 1935, and whose work touched many different branches of the tree of mathematics, including abstract algebra, geometry, and dynamical systems. One of the most profound theorems she proved (actually two with her name) is now known as Noether’s Theorem. What Noether’s (first) Theorem says is that for any conservation Law (such as energy, momentum, charge, etc), there is a fundamental geometric symmetry in the universe that corresponds to it. To express this poetically, Emmy proved that in mathematical physics, Truth (Law) is Beauty (Symmetry). Emmy’s Theorem resolved questions that Einstein had not been able to solve (!), and Einstein lobbied with Göttingen University (where she worked without pay or title) to promote her to a professorship. Eventually she was made professor, but with the rise of Nazi Germany soon had to leave the country for the US, due to her Jewish ancestry.

Sofie Kovalevskaya

Sofia “Sofie” Kovalevskaya lived from 1850 to 1891, and like Emmy Noether made substantial contributions to mathematical physics. She was a true pioneer, the first European female to earn a PhD in modern times. Together with Augustin Cauchy, she proved the Cauchy-Kovalevskaya Theorem, regarding the solutions to many equations in physics, especially those governing waves (light waves, sound waves, matter waves etc). Without her work I likely would not have had a job. Sofie was good in math but unlucky in love, her heart often broken. She had married and had children early on, and occasional star-crossed relationships later, but was also an early radical feminist and maintained a close and possibly romantic relationship with playwright Anne Charlotte Edgren-Leffler, the sister of Gosta Mittag-Leffler.  Besides her main theorem, she was also the discoverer of what is now called the Kovalevskaya Top, an exact solution to a spinning top that completed work begun long ago by Euler and Lagrange. She was also a writer, and wrote “Nihilist Girl”, a semi-autobiographical work.

“It is impossible to be a mathematician without being a poet in soul.”

 

–Sofie Kovalevskaya.

Florence Nightingale

(Yes that Florence Nightingale)

Diagram of Causes of Mortality (click to enlarge)

 

Besides being the founder of modern Nursing, Florence Nightingale had a knack for mathematics and especially statistics, and made great contributions in the visual display of quantitative information, a field which later was made popular by Edward Tufte in his seminal works. Ms. Nightingale was one of the first to make use of the Pie Chart, making clear causes and relationships in mortality among WWI soldiers.

 

Hypatia

There are so many others, such as Maria Gaetana Agnesi (the first woman appointed as full professor, but who died and like Mozart was buried in a pauper’s grave) but on my short list I have saved Hypatia for last. No likeness has ever been found, but she was said to be as beautiful as she was smart.

The first documented female mathematician, Hypatia lived in 400 AD in Alexandria, and is considered by many to be the patron saint of mathematics. And a martyr. She was the daughter of the mathematician Theon, and inherited from him the position of Director of the Library of Alexandria, the ancient repository of world knowledge. Though Theon was considered a great geometer and wrote many treatises on Euclid, Hypatia was said to have surpassed her father in mathematics and astronomy, made astrolabes, and wrote many other works and commentaries on geometry.

None of Hypatia’s works have survived, nor much of Library, whose destruction was considered one of the great tragedies in intellectual history. Hypatia was brutally assassinated by christian extremists, opposed to the “practice of sorcery, witchcraft, and mathematics”. Ironically, she was also a great teacher, and one of her most devoted students was Synesius, who studied under Hypatia as a neoplatonist, but eventually he converted to christianity and became a bishop, and contributed to the understanding of the doctrine of the Trinity.

Hypatia fought hard to save the Library, but the world was changing and she could not stop it. So much was lost when the Library fell. With it was lost much knowledge, and science, and wisdom, that we will never recover. The fall of the Library presaged the Dark Ages. Had the Library stood, some have said, we might have landed on the moon in 1492, not just Florida.

To be a woman. To be a scientist. To be a mathematician. All these things require more of one than any of us could ever know.

Be brave, these women tell us.

Be brave.

 

You Will Be Remembered

On my Sunday training runs for the Zion Half Marathon, I usually go for a run up Utah SR 9, heading towards Zion from the town of Virgin. A few miles up, I pass by this gravestone just off the highway, the only remaining thing of the ghost town called Duncan’s retreat.

I have never seen anybody stop here. Most people are tourists going 70 along this lonely stretch of highway, hell bent for Springdale lodging, and if they were looking anywhere it would be the other way, towards the Virgin River. A rock quarry lies just behind it, and nothing that would draw your attention to it.

The history book has this to say about the long-gone town:

A man named Chapman Duncan settled here in 1861. Shortly after several other families moved here also. In 1862 the Virgin river flooded and destroyed most the town. A lot of the people moved away but new settlers came. By January of 1863 about 70 people lived here. In 1863 a post office was built, a school was built in 1864 along with a meeting house. In 1866 floods took its toll on this town also and over the next few years high water from the Virgin river destroyed the fields and killed the town. By 1891 the town was deserted. All that remains of the town today is a grave of a lady named Nancy Ferguson Ott who died here in town in 1863. Her grave is located on the north side of highway 9. (Submitted by Bob Bezzant.)

My understanding from other locals is that Duncan used to live in our town of Virgin (now population 600), but “retreated” to this place far from Virgin to get away from the bustling metropolis.

When Nancy Ferguson Ott died, Duncan’s Retreat had 70 citizens, a post office, families, friends. It was a town, and they thought it would stick around long enough for there to be a cemetery with more than just Nancy, but Nature and circumstance had other ideas.

Now she is utterly alone, on a lonely stretch of anonymous highway, with no one around, no town, no post office, no friends no family.

I stopped to look at the grave marker. There are fresh flowers, stuffed animals, cards. They are replaced regularly by somebody somewhere. I do not know who.

I take some comfort in this, and I hope you do too. No matter who you are, or how alone you may feel in this world, Nancy Ferguson Ott is here, on State Route 9, miles from anywhere and anyone, to deliver this message to you: No one dies alone. You will be Loved, You will be Remembered.

Relativity, Simplified. Really.

I’ll make this (mercifully) short.

Consider this statement, which I call Ω:

Everything travels through the Cosmos at exactly the speed of light.

Ω is Relativity. That’s it. Really, it is. And Ω is true not only for Special Relativity, but the General one too. All you have to know is that the “Cosmos” means both space and time.

If you were to ask somebody what Relativity says, they would probably say something like “everything is relative.” The problem with that is, it’s not precise. In fact, it’s not even true. Some things in physics (and the world) are thought to be absolute. So I’ve been trying to come up with a good version of Relativity that can be justified mathematically, and that little box is what I came up with. If you understand what each word means exactly, everything follows from this statement, and if you like you can stop reading now and get on with life. Because that really is what relativity says.

The rest of this short post is just me rambling about why I like it. In a later post I’ll defend it. Also, here is a cool picture of a galaxy for no reason. But it’s cool (*).

Why I like This Version of Relativity

One of the things I like about this version (Ω) is that it is simple and precise, sounds a little strange — but just enough strange to be right — and answers immediately a number of other questions people ask about relativity, matter, speeds etc. For example:

Q: Will we ever be able to go faster than light?

A: NO. The reason you can’t go faster than the speed of light is that you can’t go any slower either. Nothing in the universe can go any speed in spacetime but c, the speed of light. The only thing you can ever change is the direction in space-time that you are going.

Q: Why does Relativity say that when you go faster, time slows down?

A: Because you are always going exactly at the speed c, so if you go faster in the space directions, you have to go slower in the time direction so it still adds up to exactly c.

See?  Details later, film (and math) at 11. But really, it is true. Ask a physicist. He’ll scratch his head, then say yeah, that works, then go have a beer.

(*) Photo: Centaurus A, (Image Credit: X-ray: NASA/CXC/CfA/R.Kraft et al.; Submillimeter: MPIfR/ESO/APEX/A.Weiss et al.; Optical: ESO/WFI)

 

An Anti-Christmas Carol

Another useless parasite.

Tomorrow will be June 25, the orbital opposite of December 25, and making us nearly as far as possible astronomically from the spirit of Christmas as we can be. Seems appropriate then, with the current sentiment pervading our leadership these days regarding the poor, the young, the sick and the elderly, and the changes proposed to the way our society treats these people. It therefore seems appropriate to publish here an excerpt from Dicken’s classic story, reflecting what seems to be the mood of the times, which is to tell Tiny Tim that he is a freeloader, and needs to pull himself up by his bootstraps and get a job. Or just go away and die.

Merry Christmas, Everyone.

A Christmas Carol – Marley’s Ghost

“At this festive season of the year, Mr. Scrooge,” said the gentleman, taking up a pen, “it is more than usually desirable that we should make some slight provision for the Poor and destitute, who suffer greatly at the present time. Many thousands are in want of common necessaries; hundreds of thousands are in want of common comforts, sir.”
“Are there no prisons?” asked Scrooge.
“Plenty of prisons,” said the gentleman, laying down the pen again.
“And the Union workhouses?” demanded Scrooge. “Are they still in operation?”
“They are. Still,” returned the gentleman, “I wish I could say they were not.”
“The Treadmill and the Poor Law are in full vigour, then?” said Scrooge.
“Both very busy, sir.”
“Oh! I was afraid, from what you said at first, that something had occurred to stop them in their useful course,” said Scrooge. “I’m very glad to hear it.”
“Under the impression that they scarcely furnish Christian cheer of mind or body to the multitude,” returned the gentleman, “a few of us are endeavouring to raise a fund to buy the Poor some meat and drink, and means of warmth. We choose this time, because it is a time, of all others, when Want is keenly felt, and Abundance rejoices. What shall I put you down for?”
“Nothing!” Scrooge replied.
“You wish to be anonymous?”
“I wish to be left alone,” said Scrooge. “Since you ask me what I wish, gentlemen, that is my answer. I don’t make merry myself at Christmas and I can’t afford to make idle people merry. I help to support the establishments I have mentioned—they cost enough; and those who are badly off must go there.”
“Many can’t go there; and many would rather die.”
“If they would rather die,” said Scrooge, “they had better do it, and decrease the surplus population. Besides—excuse me—I don’t know that.”
“But you might know it,” observed the gentleman.
“It’s not my business,” Scrooge returned. “It’s enough for a man to understand his own business, and not to interfere with other people’s. Mine occupies me constantly. Good afternoon, gentlemen!”


 

Automotive Math


Calculus is taught all wrong and too late. The basics could be taught in the car on the way to Kindergarten.

Here is how it goes. Any kid who’s been in a car knows that the speedometer tells you how fast you are going, and the numbers in the odometer (mileage) tells you how far you’ve gone. Another way to say that is that the speedometer shows how quickly the odometer is changing. And another way to say that is that the speedometer is the “differential” of the odometer. To be fancy, we can use a small “∂” for differential and so we say:

 

$$Speedometer = ∂\,{Odometer}$$

 

That’s differential calculus. How things change. Subtraction.

Put another way again, the odometer tells you what the Sum total distance was after going all those speeds that the speedometer indicated. Instead of saying “Sum” with a big S we stretch it out into a long skinny S like this: $\int$ and we say

$$Odometer = \int { Speedometer }  $$

That’s integral calculus. How changes accumulate. Addition.

Subtraction undoes addition. That is the Fundamental Theorem of Calculus. Duh. Next stop, Rocket Science !

THE END.

On Leadership and the President as CEO

Editor Comment: I wrote this piece as a draft  way back in 2012, during which time the Presidential CEO contender was Mitt Romney, and some had advocated for him (as for Ross Perot) because of his business experience. I never published this piece then, but the words seem even more salient, now that we are in this dubious boat. I therefore present and publish it today, in its original form. You be the judge. -NR


There has been some discussion in the course of the recent [2012] presidential campaign, with the general implication being that the President of the United states is (or should be evaluated as) the CEO of the country, and at any rate must provide leadership in how the country is run.

We are all story tellers. And the archetype of a “Leader” is a profound story, and not always a healthy one in a civilized world. In another essay, I hope sometime to discuss the two metaphorical and psychological ways I observe groups of people organize themselves and others, called Vertical (top-dog, hierarchical, canine, Yang), and Horizontal (egalitarian, all equal, “it takes a village”, feline, Yin). A Leader is a character in the “Vertical” view of the world. I’m not saying either view is entirely good, but each has its hazards.

I tend to agree with the founders of this country who had grown sick of Old World countries that were “run” by somebody, whether their name be King George or Putin or Berlusconi…or Obama if power ever goes to his head. Regardless of how my philosophy has evolved, to this day I remain a profound admirer of Ayn Rand, whose supreme heroes John Galt and Hank Rearden were brilliant, rational and secular men who did not seek leaders and did not seek to be a leader — and all they asked of the government was for it to Get The Hell Out of Their Way and let them pursue their work and happiness, to the best of their ability.

Galt and Rearden are heroes for (as Obama recently said in Rolling Stone) smart 17 year old intravert misfit teenagers, who feel the world does not understand them. And remain so for those who grow up to be smart adult emotionally integrated intravert misfits who name their houses “Anthem” for more reason than one.

Regarding CEO’s, an article from Thomson Reuters’s archives indicates that the three top skills for an entrepreneurial CEO are:

  • Financial Management
  • Communication
  • Motivation of Others

Good god, I must waste half my time at work convincing whoever is my current boss not to promote me to some “leadership” or “management” role involving people skills and such. Communication I can handle, preferably by email and preferably from far away here in Zion Canyon; if they really want eye-contact we can Skype. I’m Scotty in the engine room and dilithium crystals are fragile, let Kirk deal with the damn aliens. I have no interest in learning motivational psychology or finance professionally. But I know myself. I remain a shy but bright mathematician and software architect, who solves difficult problems in the structure of programs and languages that make it possible for those who write the higher-level business logic to express it coherently, and execute it fast and effectively. For me to be promoted to an executive or managerial role would be tragic for all involved, and I state as much explicitly in my annual employee performance review. Just get the hell out of my way and let me do work that makes me happy and makes you a profit. Everybody wins!

The genius of our country is that it was the first on the planet to be designed based on principles, predicated on the idea that its purpose was to establish ground rules preserving the rights of man, and as men are prone to abrogate power, the founders split the government and its powers into three separate branches. Of these three, only congress is charged with the formulation and passage of laws and its leadership role is composed of many men and women to prevent a single ambitious man dominating. The executive branch was originally intended to be purely executive and the president was intentionally weak, and though he has veto power, certainly not a leadership role. Indeed, George Washington refused proposals that he be made King.

The “leaders” of our country were always reluctant ones, and rightfully so.

Alas, as time has passed, the nature of humanity seems to be that some segment who leans toward “The Vertical”  will always seek a strong powerful figurehead to guide them and save them, whether in the form of an all powerful deity, or as a testosterone laden bully/hero such as Mussolini or Stalin (or even FDR, when he tried to pack the Supreme Court). And unfortunately, with a strong powerful leader, you also wind up with a large, powerful government.

I still agree with fiscal conservatives that the proportion of GDP consumed by the current government is far too large, and poses an existential threat to the country. One place of many where I disagree with Ayn Rand and Libertarians now is on the minimal “ground rules” established by a morally defensible government: For example, I would expand the Libertarian “national defense” segment to include defense against natural disasters and disease, from which an argument for a universal baseline healthcare system can be derived, and by my thinking should come directly out of the Defense Budget.

Getting back to leadership, I can only think of a few times where the person in the role of the President has stood up and used the “bully pulpit” to deliver helpful words crafted in thought and grace, to speak to the American people in a way that either united them all as a people, or brought a modicum of closure. FDR’s fireside chats, JFK’s moon speech, and yes Reagan’s address following the Challenger disaster. But there were others such as Martin Luther King Jr, who also served that role, armed only with the power of their own spiritual substance and the confidence in their cause, with no need of elected office.

If we are talking about the president leading the country by putting in place dramatic changes in the economic and regulator power of the state as well as its size, then there are only a few examples of this, one with FDR (who dramatically increased the size and scope) and Reagan (who dramatically decreased it). As president, however, both remained powerless to implement their goal without the fact that congress also had to be in the same party in order to propose and pass those laws, as well as approve members of the Supreme Court to interpret  and not strike down those laws. In general, however, it has been observed  that Americans seem to be more comfortable when the two branches that pass and sign bills are split. Americans don’t seem to care so much who is in which branch, just so long as they are at odds, in line with the “checks and balances” concept.

We are Americans, where each person is the owner of their own destiny, and who seem to mistrust any one man or party to claim power over them, even if we may agree with them. The president’s activities by rights and by the founder’s intent should be as much in the daily news as the day to day activities of the school janitor. The fact that some have come to seek a President to act as a a leader and CEO of the country is a disturbing state of affairs, in my opinion.

I am *Terribly* Sorry to Hear That

This micro-post serves as a permanent placeholder for the coming years, in which those who voted for this administration will I suspect find that many promises have been broken. And after the first days of triumphant fanfare and euphoria, unfolding events both sad and tragic will be all too common, and not at all what they were promised by shady and cynical salesmen.

And so, in the face of such monumentally bad omens, saying “I TOLD YOU SO” seems too much like inappropriate gloating, and the wrong attitude to take in a world that will already be suffering under the weight of a bloated and narcissistic ego.

And so, in place of that response, I will say in a spirit of compassion the following words, exactly the same each time, with two *asterisks* to emphasize the terrible, and to point back to the ALL CAPS statement above which I am thinking, but will not speak:

I am *terribly* sorry to hear that.
My one promise is that those words will be sincere, and I truly will be very sorry to hear about whatever new travesty has just ocurred. That is my New Years resolution, such as it is.

From Euclid to Euler to Einstein

It is of course the holiday tradition this time of year, to exchange gifts and ponder over how you would explain modern mathematics to the ancient Greeks.

In line with the latter part of that tradition, I’ve been sketching out a diagram to explain Euler’s number $e$ (2.71828…) to Euclid. It turns out that even though the classic Greek mathematicians knew all about the number π (3.1415…), they never knew about or defined the number $e$. Which is a shame, because they could have. And had they done so, they could have beaten Einstein to the punch 2500 years earlier.

Just a quick note here: for those of you who have not heard of $e$, it pops up all over the place in science, and especially when things are growing or accelerating. For example, suppose you just crossed the state line, and for some reason you thought that the mile-markers were actually speed limits, so that at the one-mile marker you slowed down to go at one mile an hour, and so on. Suppose that there were a lot of mile markers along the way, and so you were continuously speeding up with each marker. Obviously you would be going pretty slow, but at least you are speeding up. It turns out that if you obeyed the signs to the letter, by the end of one hour from mile marker one you will be at the $e$ mile marker, and would be going $e$ miles an hour.

In any case, after much fiddling around and fanfare, here is the diagram I came up with that I think would make Euclid happy. It is a “proof without narrative”, and simply uses the classically understood conic sections (e.g. circles, and hyperbolas) to show how the numbers π and $e$ may be used to relate areas of pie-shaped sectors in two conic sections, to the linear measurements along their respective curves:

One of the things I like about this diagram is that on the one hand it shows how these two numbers are similar, in that they both provide a ratio relating the area of a sector in each type of conic section, with a linear measure, but on the other, we see how these two numbers differ in a fundamental way with successive sectors.

For circles of radius 1, its area compares with its radius squared by a ratio of π (so the pie-slices are each π/8). For the hyperbola, drawing a line from the center to the vertex of the hyperbola, a sector of area one is made by drawing a second line whose x-axis length differs from the area by a ratio of $e$. In both cases we have a ratio relating a linear measure to an area.

But at this point the similarity ends. For as we go to successive circular arcs, the areas remain in fixed linear ratios, so to produce a quarter of a circle, you have an arc-length of π/4, and so on. But for the hyperbola, to produce a sector of area 2, you need to draw a line segments whose x-axis length is not $2 * e$, but $e$ to the power of 2, in other words $e^2$. For an area of three, you need to use $e^3$, and so on.

So what we see is that the number π seems to be most commonly used as a linear factor or ratio, having to do with rotational symmetry in space, while the number $e$ seems to be used as the base of an exponent, and is involved with things that grow exponentially over time.

Which brings us to light, waves, and Einstein’s space-time.

What do cones, planes and conic sections have to do with spacetime? Suppose you turn a flashlight on and off quickly. The light pulse from that event travels out in all directions at the same speed, $c$, the speed of light. Einstein (and Minkowski) suggested that we view the event where time plays the role of a fourth dimension. If we toss out one of our three dimensions, and make the time dimension the z-axis, we can visualize the light propagating out.

So in the picture on the right, the horizontal plane represents space at time $t=0$, and the vertical dimension is time, with the “up” direction representing the future, and “down” representing the past. The flashlight has just gone off at time zero, but now the light wave is expanding out in a circle, getting larger with time. And so as it grows over (upward) time, the expanding circular wave traces out the “future light cone”. Conversely, all of the light from the past that reaches us can only come from the region below the plane, marked by the “past light cone”.

The thing to note is that these “space-like” planes are always horizontal, though they may tilt a little due to relativistic motion of the observer. Space-like planes can be identified by the fact that their “normal” line (the one perpendicular to the plane) are pointing roughly up, in a time-like direction. Space-like planes can only intersect light-cones in circles or ellipses. In no case can an observer’s “plane” ever become vertical, so that its normal vector is pointing in a space-like direction outside of the light cone. Such planes are called “time-like”, and have the property that they always intersect light cones in hyperbolas.

So I am hoping that you are starting to see how I think these two numbers $pi$ and $e$ are related, but also very different. Somehow, the number $pi$ is related more to space, and to circular rotation in space, while $e$ seems to be related to time, hyperbolic curves, and exponential growth over time.

It turns out that we can even be very specific about how $e$ and $pi$ are related to each other, but it requires the introduction of a number that the ancient Greeks would have no concept of, and that is the number $i$, the square root of negative one.

The relationship was itself discovered by Euler himself, and has come to be known as Euler’s Equation, and has also been called (at least by mathematicians), The Most Beautiful Equation in the World. I hope some time in a future post to try to explain what the equation means, but for the moment, we will just display it here and be done with it.

$$ e^{i\pi} + 1 = 0  $$

And yes, this is how I spend my holiday vacations. Having Fun ! Happy new year !

 

The Geometry of Meteor Showers

Whenever a meteor shower is coming up, the news gives details on how to find the constellation in which the “radiant” can be found. Don’t bother trying to find the constellation. Too much work. Here is all you really need to do:

On the night of the shower, go outside around 2am. Look eastward, toward where the sun has been rising, and halfway up the sky, along the path the sun takes. That's the center ('the radiant'). Further away from this point the meteor trails will be longer.

That’s it. The rest of this post is just my rambling about the geometry (or astrometry as it were) that makes this all work. You won’t need it. If you come out sooner, around midnight, the radiant will be close to the horizon, and as it gets closer to sunrise the radiant will be almost overhead.

The Radiant

If you study the pattern of meteors in the picture above, it looks like we are flying through a bunch of stars very quickly, and that the center point where all those stars appear to be streaking from is simply the direction that we are flying.

It turns out that is exactly what you are seeing. The center point (in the upper left quadrant of the picture) is called the Radiant of the meteor shower, and it is the current direction in which the earth is moving, as it travels along its orbit around the Sun.

The Picture

Here is a (simplified) picture describing the general situation. To keep things simple, I’ve put the little guy (who’s supposed to be us) right on the earth’s equator, around 3am his time. We are looking down at the earth from above the North Pole, and the earth is rotating counter-clockwise on its axis. Meanwhile, the earth is travelling around the sun at 18.6 miles a second, going from right to left in the picture.

 

meteor_diagram

The comet dust in the picture was left behind by a comet years before, and now is for the most part not moving much. The earth however is plowing through the dust trail at 18.6 miles/second, and so the relative motion of the dust to the observer is likewise 18.6 miles a second, or about 30km/s.

That speed, by the way, adds a lot of energy to the situation. Many of the comet dust particles are small, some just grains of sand. But if we take a quarter-inch piece of iron, with a mass of one gram say, and compute its kinetic energy when the earth hits it, we get

$$E = \frac{1}{2}mv^2 =\frac{1}{2}(1gm)(30km/s)^2 = 450,000  Joules$$

Now a Joule is the amount of Energy to drive a one Watt light bulb for a second, which is about how long a meteor flare takes. So the light that our quarter inch piece of iron is putting out during that second is close to a half a megawatt of power. Impressive.

Another Picture Closer In

So here is a much closer picture. We’ve now rotated the picture so that the little guy is on “horizontal” ground, and we only see a small slightly curved part of the earth. The atmosphere is a very thin shell not more than 70 miles above the earth (1 percent of the earth’s diameter), and the shaded part is what the little guy can see from where he’s standing. It is a flat lens shaped piece of atmosphere, and the comet dust is coming in at about a 45 degree angle, about to slam into that circular lens. I’ve drawn a cylinder around all of the dust that will hit the part of the sky that the guy can see.

meteor_detail

Now if you look at the cylinder of comet dust coming at you from the little guy’s perspective, the rays of dust look like this:

cylinder2

Which looks just like the photo of the Geminid meteor shower. So, the reason that showers look like rays flying away from the Radiant is simply a matter of perspective, and the Radiant itself is just the direction that we are are flying, along earth’s orbit.

A Bigger Picture, Further Out

Just to tie everything together, here is a diagram showing the geometry of a typical meteor shower, arising from a regularly reappearing comet such as Halley’s comet:

meteor_shower_big_picture

 

In the case of Halley’s comet, the diagram shows how the orbit of the comet may intersect the Earth’s orbit in two places. In the current picture, the Earth is passing through one of the intersections, and is going in the direction of the constellation Orion (bottom left). This is the Orionid meteor shower, which this year (2016) will be visible from October 2 to November 7. The other intersection occurs when the Earth is heading in the direction of Aquarius, which happens around May 5-6, during the Eta Aquarid’s meteor shower. Not all comets have orbits which intersect Earth’s orbit twice, but Halley’s does.

Valid Religion

This monastery was just down the street from us when we lived in Long Beach ten years ago. It has a beautiful view of the ocean and we’d often pass it while out for a walk along the beach. To the left you can see a white shrine in which the Virgin Mary is standing. It is a very pretty installation though we sometimes would call it “Virgin Mary on the Half-shell.” About half the time we’d pass by, there would be someone there, praying to the virgin, or lighting candles or incense.

The monastery used to be a convent owned by the Catholic Church, but was long ago sold to a Vietnamese Buddhist sect, and they have converted the building into a monastery for their monks. Every so often you would see one of the monks leave the building in their saffron robes, but for the most part they keep to themselves and maintain a quiet life of contemplation.

virgin_maryNow here’s the thing: not only did the monks not tear down the Virgin Mary shrine, but they actively keep it intact, tending to the plants around it, the benches were people come to pray, and bring fresh flowers. I have never asked the monks about why they do this, or if it was a condition of the sale, but they never seem to mind. The Buddhist tradition maintains a very ecumenical respect for the beliefs of others, and maintains that there are many paths to enlightenment.

In my personal system of philosophy, I have a name for religions of this type: VALID.

I have to explain what I mean by this. With the possible exception of mathematics, I doubt that any statements of a philosophical nature will or ever can be determined to be TRUE. Thus, it is in my opinion foolish for any religion to declare itself True, as it is the height of arrogance, and proclaims that all of the other fifty thousand religions in the world are False. So, what I mean by a VALID religion is one in which the religion makes an admission of the following things:

  1. Fallibility, that this religion may have made some mistakes
  2. That absolute Truth is unknowable by man,
  3. That other religions may have important and valid points, and be a legitimate source of hope and inspiration in the lives of its believers, and rightfully so,

A VALID religion is therefore one in which it recognizes other religions as potentially equally valid, and there is no Law of Excluded Middle (as with True/False) that gets in the way.

Any religion which declares itself to be the sole possessor of this elusive thing called The Truth, is by my definition INVALID. Alas, this includes almost all religions of the world, including many brands of atheism.

This monastery is, then, to me a triple-shrine. First, to the Virgin Mary herself, secondly to the buddhist monks that respect people’s belief in her, and third to the possibility that a religion could if it so chooses become VALID — a thing which seems to me to be the only hope for peace among humans who choose to believe in the wildest of fairy tales. And to date, this monastery represents the only example to my knowledge of a valid religion.

Rainbow, Part II: Yellow Is An Idea

This is part II of my discussion of color which began with Part I, “The Infinite Piano”. In the first part I explained that the colors of the rainbow are single “notes” on an infinite piano whose keys are pure “tones” of light, and the “sheet music” for a more complex color such as PINK can be written as a 3-note chord composition in RED, GREEN, and BLUE. This composition can be written out over the color piano keyboard with three vertical bars, each indicating the loudness or softness of each of the three keys we need to play, using ranges from 0 to 255, like this:

pink3

We can further shorten this musical notation by saying (Red,Green,Blue) = (255, 192, 203). Now you may think that I just made up those particular numbers, but in fact if you check with Wikipedia, the internet standard for color on computer displays has exactly these three values for the color pink. They chose the range 0 to 255 because it is easy to express using 8 bits — which makes computers happy.

We live in the computer age, and this (R,G,B) system is now used to define all the colors that you can see on a computer monitor. So, it sounds like color is three dimensional, and you can represent any color in nature (or at least in a photo of nature) using just three colors. But is this true ?

Anyone who has tried to match paint colors may doubt this. Each paint manufacturer has their own system of specifying colors, and complex formulas of mixing their “component” pigments into Salmon, Chestnut, or other copyrighted name and color. There are many systems of defining color, such as Munsell and CIE-Lab, which are 3-dimensional, like this:

By SharkD - Own work, CC BY-SA 3.0, https://commons.wikimedia.org/w/index.php?curid=8401562

3D Munsell color space (Wikipedia – credit)

These systems are oriented toward luminance-based applications such as TV’s and computer monitors that emit their own light. There are also CMYK (Cyan-Magenta-Yellow-BlackKey) and Pantone™ systems, which are effectively 4 dimensional dimensional and used mostly in pigment-based applications such as printing and paint. But Pantone also had a six-dimensional version called Hexachrome, which add Orange and Green to form a CMYKOG space (now discontinued), and there is also a CcMmYK system used in six-color inkjet printers. These latter are called “subtractive” systems, because the pigments effectively absorb colors from white light to give you their indicated color.

So clearly something must be going on. Why do we even think color is three dimensional, when there are so many color systems using more than three. What’s up?

The Yellow That Isn’t There

Let’s take a closer look at this Wikipedia computer color thing. If you look up Yellow in wikipedia, you’ll see that standard Yellow is defined in color coordinates by (R,G,B) = (255, 255, 0). But if we plot that “musical chord” out on our piano we get this:

yellow2

Now this is crazy, because there is clearly a “yellow” key halfway between green and red, and we aren’t hitting that key at all. Instead we are leaning with a strong 255 “forte” on both RED and GREEN. Indeed, in the same Wikipedia entry for Yellow, it indicates that the “spectral” coordinates of Yellow is 570–590 nanometers. This is the wavelength of the light which is colored yellow in the rainbow spectrum.

To understand what is going on requires an understanding of human beings more than the color spectrum, and how we evolved. Modern humans perceive color with the use of three kinds of cells in the retina of our eyes, called cones. These cones come in three types, each of which respond only to specific “chords” in the color spectrum. The three chords look something like this (approximately):

cone_spectrum_2What this says is that we have in our eyes three kinds of cells (not counting rods which detect brightness), which respond to “color chords” that are centered (roughly) around the blue, green and red keys. There is no cell that responds just to “yellow” chords, and so the way that we “see” the yellow color is that our brains get strong positive signals from both the Green and the Red cones.

One of the interesting consequences is that it is possible to make a person “see” yellow even if there is no yellow in the light at all. All you have to do is to take a pure green and red light (such as from two distinct lasers), and shine them on the same spot on the wall:

red_green_yellow

Our retinas will report to the brain that where they intersect it is getting a strong green and red signal, and the brain will interpret that as yellow — even though a light spectrometer pointed at the wall will report that there is no yellow there at all. It is a color optical illusion !

Here is the take-away from all this: the color YELLOW is an IDEA, as are all other colors. It is something unique that our brain thinks — a state of mind — in response to what the outside world is doing. In the case above, the YELLOW our brain “sees” is entirely in our own heads. Now most of the time, in nature, there really is a yellow frequency light wave “out there”, and we know from the yellow in the rainbow that this frequency of light actually exists. You can create a pure yellow by simply dropping salt into a flame (sodium ions radiate at that color). But the idea of yellow must be distinguished from the light that usually triggers it.

And so, YELLOW as a specific color of light must be understood as a separate dimension from RED, GREEN, and BLUE. So how many dimensions does color really have? We will explore this further in the next post, “Shadows of The Infinite.”

Remembering Paul Sidney

At various times from fourth grade through eighth, Paul Sidney was my best friend and worst enemy. I have now lived for fifty seven years, and Paul retains a special place of honor, being the only person on the planet that I have ever punched in the nose (or wanted to).

That was in the sixth grade, at Steven Millard Elementary school. I can’t even remember exactly what it was about, though Paul did have a biting wit and what we would now call a “snarky” attitude. Very likely it was a sarcastic comment he made at the time about a crush I had on Diane, a girl I first met in square dancing class in fourth grade. Now that I think of it this was Paul’s great talent, being one of the few people in whom I felt I could confide my deepest feelings, and who later would use those secrets to torment me in artful and insidious ways.

It has taken four years for the news to reach me that Paul had died, June 12, 2011. He was 53.

I had always thought that I would be hearing about Paul, over the years. He was a very good writer in Junior high school, and we had something of a rivalry in creative writing. He could have been a writer, or an actor, graphic artist, or any number of things. I googled his name every so often, looking for books published, lectures given, organizations he had founded, Tony-award winning musicals starring Nathan Lane written by him. Nothing. Somehow he had just fallen off of the map.

The obituary was just a note, no detail, no evidence of a memorial with thousands of friends and admirers, remembering him, telling stories, laughing, crying, people who were touched by him.

Eleanor Rigby, died in the church
And was buried along with her name
Nobody came

After eighth grade, I and most of my classmates went to Irvington High School. Paul did not. For reasons we never learned, he went to Moreau, a Catholic High School in Hayward. I only saw Paul twice after that. Once was at the Fremont Main Library by Lake Elizabeth, and he was studying at a table, probably for a class. We said hi. The only other time was a few years later, my senior year, when I saw that he was appearing in a Halloween stage production of “Dracula”. He played Beddoes, the assistant.

That seemed appropriate. Paul always had a sense of the macabre. I remember hanging out with him, reading his creepy comic books, graphic novel versions of Edgar Allen Poe, “The Telltale Heart,” complete with gruesome beating hearts.

For some reason Paul always reminded me of Lucy van Pelt, from Peanuts. Smart beyond their years, a bit crabby, a fussbudget, but with an acerbic wit, a sabre that he could unsheathe at the drop of a malapropism. Something feminine or feline as well. Even this seventh grade photo (above) has him sporting a faux leopard-skin vest. Oscar Wilde.

Reading back on this piece, it almost sounds like our relationship was romantic, a love-hate thing, doesn’t it? I don’t know, I was a kid, and pretty much clueless. All I knew was that he was a very smart guy, and one of the few who challenged me in the world of ideas, and words. Perhaps I did love him.

Paul and his sister Kay, 2000

Paul and his sister Kay, 2000

I contacted Paul’s sister Kay, and wrote a letter (on paper, with pen), asking about Paul, and what happened. I wish I could say that I was surprised, but was not. Things did not go well for Paul. His parents divorced, and in High School Paul began to exhibit the first signs of Schizophrenia, a disease with which he struggled the rest of his life. His family tried to help him, but it is in the nature of the disease that having any sort of life as I would have wished for him is virtually impossible. He ultimately died from the effects of COPD, a congestive lung disease exacerbated by a lifetime of smoking.

As I once wrote, a small mouse in Connecticut once taught me that the greatest gift that you can give someone, is to remember them. Each life, no matter how small, touches someone. Their life matters. They had a life, they had a story.

This one is for you, Paul.

A Thousand Cranes

origami crane

I will be giving lessons on how to fold these cranes at the Future Faire in Virgin on June 20. Feel free to come down and join us, or fold your own crane at home (directions below) and send them to me, at the address in the instructions. Read on for the story…

Our little town of Virgin has been through some divisive troubles lately, and there are some hard feelings going around. At heart though, I believe that everybody in this town cares about Virgin, whether they are sons of pioneers or newcomers from out of state, or upstream refugees from bustling metropolitan Springdale. I am sure they all have only the greatest hopes for it to become the town of their dreams.

When I was in fourth grade our teacher, Mister Haney, taught us about Japan, and the arts of calligraphy and origami. There is an ancient legend, he told us, that if you fold a thousand origami cranes (Senbazuru) you will get your wish and prosperity will be yours for a thousand years.

I am announcing the “Thousand Cranes for Virgin” Project. I am asking for anybody that cares about our town to take a few minutes and some origami paper, learn how to fold a classic origami crane, and send them to me. I hope that within one year, we will have over a thousand cranes, which we can string together and keep in our new Community Center (the old restored church), as a sign of our common love for this town, and to ensure its happiness and prosperity for a thousand years.

Here’s how to fold an origami crane, and where to send it:

crane

The Ancestor’s Tale

Summary

ancestor

If you are not in the mood for my idle chatter to follow, here is the bottom line: I love this book. Whether you buy into evolution or not, everyone should read “The Ancestor’s Tale” because it is a marvelous bit of writing which will challenge you to think and rethink the surprising realities and consequences of your own position. Even if you completely buy into evolution, this book dares you to accept its disturbing implications, as unnerving as they may be. And that is always a good thing.

But First, a Story

In the Gene Kelly movie “Singing in the Rain”, Don Lockwood (played by Gene Kelly) and Lina Lamont (played by Jean Hagen) are silent-era movie stars about to make their break into the “talkies” with a musical. The only catch is, Lina can’t sing and has an awful accent. So they invent the idea of “dubbing,” and have Kathy Selden (played by Debbie Reynolds) to do Lina’s parts while Lina is lip-syncing. So, in the movie, what you see is Kathy talking and singing in a beautiful deep voice behind the stage, while Lina is mouthing in front of the camera.

lina-7

Now here’s the lesser-known fact: in reality, Debbie Reynolds did not sing in any of those parts — she has a midwestern twang. So they needed to find somebody who could do the voice-overs for Debbie Reynolds doing voice-overs for Jean Hagen’s character.

And here’s the best part: you know who they used to do the talking voice overs for Debbie Reynolds doing voice-overs for Jean Hagen?

Jean Hagen. Turns out that the awful Bronx accent Lina Lamont has is a fake accent, and that actually Jean Hagen has a great voice.

You just can’t make this stuff up.

The Writer’s Tale

No matter how wild and interesting a story a writer may concoct, it seems, Reality has a way to come up with something far more strange than the writer could ever invent. Indeed, if you were to take an insanely crazy true story and try to sell it as fiction, you would have trouble: it would be too weird to be believable. Nobody would buy it.

Irony: although men of great faith are held in high esteem when they hold fast to their beliefs, in spite of all opposing forces (including the hard evidence of scientific experiment), in fact this is one of easiest things for humans to do. In fact, all evidence appears to indicate that when confronted with incontrovertible evidence that negates a strongly-held belief, most people’s response to this is to double-down and hold even stronger to their now-disproven positions.

The scientist, in contrast, has if anything a more difficult and heroic task: and that is to be willing at a moment’s notice to discard their most cherished theories, beliefs, ideas and standards, if it is shown by the findings of experiment, peer-review, and the evidence before their own eyes, that their precious stories — however “reasonable” sounding in their ears — do not describe the real world around them and cannot be used as a guide to how the world works, and they must now embrace their opponent’s creed — the one that they had branded heresy.

The history of science is one of a never-ending series of discoveries which suggest stories that are not only stranger than anyone every imagined, they have become stranger than anyone ever could have imagined. Quantum mechanics. Relativity. Big Bangs. Black Holes. Continental Drift. And among the strangest of tales is Evolution — this latter being a story that is even more difficult to accept because of what is says about ourselves, what we are and how we think about ourselves.

The year 2014 marks the tenth anniversary of the release of Oxford professor Richard Dawkin’s magnum opus “The Ancestor’s Tale,” and it has since its publication joined my short list of books to which I have found myself returning again and again. Whether you “buy” the story of Evolution or not, this book will challenge you, all the way down to the core of your most firmly held beliefs.

Pilgrimage

Dawkin’s idea for the book is based on Chaucer’s Canterbury Tales, in which pilgrims on their way to Canterbury each tell their own stories. But now, you are the pilgrim, and your trail is your family tree, which you must follow back in time. And as you travel further and further back, you will begin to meet up with friend and neighbors, at the crossroads where you find that you both have common ancestors.

And part of the motivation for using Chaucer’s book is the remarkable calculation that implies that for any two people now living on the planet, their personal paths of pilgrimage will meet at or sooner than the 1400’s — the time during which the Canterbury Tales was written.

The name that Dawkins gives to a “common ancestor” is Concestor, and the Rendezvous or crossroads is for Concestor Zero — which is his name for a single common ancestor of All Human Kind alive today. In other words, on your personal family tree, Concestor 0 is the first ancestor you find who is also in the family tree of every single human being alive on earth today.  The fact that there is such a person in your tree is in itself one of the first remarkable facts that you are challenged with. And yet, Dawkins presents a clever proof, using only math and logic, and unarmed with the story of Adam and Eve, to argue that at least one such couple must exist in your tree that unites all of humanity. And this is what makes this book great, is that Dawkins doesn’t just claim things, he proves them. To see the proof, you will just have to read the book.

 

 

How To Go Faster Than Light

Bottom Line

For those with limited attention spans: yes, in this universe, with a powerful enough rocket you really can go anywhere in the universe as quickly as you like, in your own lifetime, without resorting to any medical tricks like suspended animation. Einstein’s theory of relativity won’t stop you from getting there, the same day even. The hard part is just getting enough energy — and working out the math.

Speed Limits

enterprise-tos

One big downer — if you can call it that — most people take away from Einstein’s theory of special relativity is that nothing can go faster than light. We are a species that likes to explore, after all, and the idea that there is a depressingly slow speed limit imposed on us by nature makes it very difficult to journey through the galaxy.

Science fiction often addresses this either by inventing a device to “warp” space-time (Star Trek), or by adding a few extra dimensions to the universe and bypassing normal space by jumping into hyperspace (Star War).

A lot of this comes, I think, from a confusion about how the universe actually works, as described by Einstein’s theory of relativity. (Note: given the recent creationist attempt to color the word “theory” as meaning something tentative, I prefer to use Richard Dawkin’s coined word “theorum” — similar to theorem — as indicating a theory that is so well established by overwhelming evidence that it might as well be an undebatable mathematical theorem).

The Confusion

Here is the deal: while it is true that to people watching from earth a spacecraft can never be observed to go faster than light, that doesn’t mean that the passengers on the spacecraft have the same experience. In fact, what Einstein’s theory would say is that as far as the passengers can tell, it seems like they can go as fast as they like. Due to the relativistic “warp” of space-time as you approach the speed of light \(c\), the passengers experience time much more slowly and their “effective” speed as they travel through space appears to be much greater than light.

Let’s do the numbers.

Some Terminology

In the “Star Trek” series they used the “warp \(N\)” terminology to refer to “effective” speeds that were \(N\) times the speed of light \(c\), so that “Warp Two” for example was twice the speed of light or \(2c\). In that series they had a special “warp drive” that bent space-time around so that they would go faster, but the actual fact is that in our everyday world just the mere act of going faster by any means actually warps space-time.

The Warp Equation

I plan on using the trekkie terminology (and standard relativity) to state and prove the following interesting fact:

The Warp Equation
If you have a payload with mass \(m_{payload}\), and a means of converting matter into kinetic energy with 100% efficiency, then the mass \(m_{fuel}\) of fuel needed for you to travel at an effective speed of Warp \(\omega\) where \(\omega > 0\) is given by$$ m_{fuel} = {\omega}^2 m_{payload} $$

 

So for example, in order to travel at Warp 2, a person of mass 80 kilograms would require 320 kilograms of (say) a proton-antiproton fuel in order to travel at that effective speed. That is roughly equivalent to 6,400 Megatons of TNT. Coincidentally, that is almost exactly the combined explosive power of all nuclear weapons now on our planet. That is a hell of a lot of energy, but the point to be made is that is within the bounds of our current technology.

The fact that you have to square the warp factor to get the amount of energy to go that speed makes perfect sense. Even in classical Newtonian physics, the energy related to going at velocity \(v\) is given by

$$E = \frac{1}{2}mv^2$$

so doubling the velocity \(v\) on the right hand side multiplies the energy by four. The fact that the energy happens to be equivalent to four times your payload’s mass comes from Einstein.

The way in which we’ll prove this is to first calculate how much matter is needed to attain an observed velocity v, and then figure out what the relationship is between the observed velocity, and what effective velocity the passenger actually experiences. Note: I have no doubt that there is probably an easier way to derive this formula. But this is the one I came up with and it isn’t all that complicated.

Conversion of Matter to Kinetic Energy

Let’s start with Einstein’s equation:

$$E=mc^2$$

What we are going to do is to use this equation, together with the law of conservation of energy, to compute how much matter it takes to accelerate a payload \(m\) to (observed) velocity \(v_{o}\). Now as the observed velocity \(v_o\) approaches the speed of light, the relativistic mass of the payload becomes:

$$m_{relative} = \frac{m_{payload}}{\sqrt{1-(\frac{v_o}{c})^2}}$$

Now Einstein’s equation for energy represents both the energy of the mass at rest, together with the (kinetic) energy of the mass in motion. And so, if this mass was put into motion by the conversion (at rest) of a certain mass \(m_{fuel}\), where

$$m_{fuel} = \alpha m_{payload},   where  \alpha > 0$$

Then since energy is conserved we can relate the conversion of the mass \(m_{fuel}\) into motion \(v_o\) by:

$$ (m_{payload}+m_{fuel})c^2 = E_{rest} = E_{moving} =\frac{m_{payload}}{\sqrt{1-(\frac{v_o}{c})^2}} c^2$$

so dividing both sides by \(m_{payload}c^2\)

$$ 1 + \alpha = \frac{1}{\sqrt{1-(\frac{v_o}{c})^2}}$$

squaring both sides and solving for \(v_o\) we get the following rule:

Matter to Velocity Conversion
For a payload of mass \(m\) and a ratio \(\alpha > 0\), if fuel \(m_{fuel}=\alpha m\) is converted to kinetic energy, the observed velocity \(v_o\) of the body will be$$v_o = (\sqrt{\frac{\alpha}{1+\alpha}})c$$

 

This jibes with what Einstein said about observed velocities, as the right hand side will never be greater than the speed of light \(c\). As the ratio \(\alpha \rightarrow \infty\), the velocity goes to \(c\), so we can get as close to \(c\) as we like — but no further.

Velocity – Observed and Effective

So now we come to the idea of “effective” velocity. The weirdness of relativity comes from the fact that as the observed  velocity \(v\) of ship approaches the speed of light, the passenger’s own time-scale is compressed by what’s called the Lorentz-FitzGerald contraction, according to the formula

$$t_{effective} = t_{observed}\sqrt{1-(\frac{v_o}{c})^2}$$

(From this point on we will just write \(t_e\) and \(t_o\) for \(t_{effective}\) and \( t_{observed}\) respectively) Then given a fixed distance \(\Delta x_o\) as measured by the observers on earth, the effective velocity as experienced by the passengers when traversing that segment of space over their time \(\Delta t_e\)  is:

$$v_e = \frac{\Delta x_o}{\Delta t_e} = \frac{\Delta x_o}{\Delta t_o\sqrt{1-(\frac{v_o}{c})^2}}$$

which in turn simplifies to this formula for converting observed to effective velocity:

Observed to Effective Velocity
$$ v_e = \frac{v_{o}}{\sqrt{1-(\frac{v_o}{c})^2}}$$

 

The important thing to note about this concept of “effective” velocity is that we are computing the ratio of observed change in our distance with our own experienced change in time. Suppose for example that over centuries the earth residents went out and placed “mile markers” along your path to your destination. Then once you got up to warp speed, not only would your sense of time be compressed, but distances that you measure out with your tape measure would also be compressed by the same Lorentz-FitzGerald contraction. Consequently, if you were going at Warp 2, for example, your sense of time elapsed going from one mile marker to the next would be cut in half, but also (this is the point) the distance between those two fixed mile markers as the earth-people laid them out would also appear to you to be half. So if you were instead to calculate your velocity by dividing your measured distance by your time, you would never get a value at or greater than c.

All Together Now

So if we start with fuel \(\alpha m\) which we use to accelerate our mass \(m\) to the observed velocity \(v_o\), we can use the two formulas we just derived to express the effective velocity \(v_e\) as a function of \(\alpha\). We can rewrite the “Matter to Velocity” formula as

$$ (\frac{v_o}{c})^2 = \frac{\alpha}{1+\alpha}$$

So our effective velocity formula simplifies the bottom of the fraction to

$$v_e = v_{o}\sqrt{1+\alpha}$$

and then substituting the formula again for  \(v_o\) we see that our fuel mass \(\alpha m\) gives us an effective velocity of:

$$v_e = c\sqrt{\alpha}$$

Thus if we have defined velocity “Warp \(\omega\)” to be \(\omega  c\), then we can write

$$\omega = v_e / c = \sqrt{\alpha}$$

So that to attain an effective velocity of Warp \(\omega\) we must use a fuel-payload ratio of \(\alpha = \omega^2\), ie

$$m_{fuel} = \omega^2 m_{payload} $$

which is exactly the “Warp Equation” we were to prove. QED

Let’s Do the Time-Warp Again

It should be pointed out that of course to the observers on earth, even though you are going at an effective speed of Warp \(\omega\) you will never appear to be going faster than \(c\) and so it will take you a long time to get where you are going. You, however, will not experience that, and so you will effectively be travelling through time much faster than your friends at home. How much faster? According to our formula above relating \(t_e\) to \(t_o\), and expressing that in terms of the warp factor \(\omega\), we can show that the time-warp you experience will be:

$$t_e = \frac{t_o}{\sqrt{1+\omega^2}}$$

And so, in our example, the 80kg person travelling at Warp 2 will feel like they’ve reached their destination in \( 1/ \sqrt{5} \) of the earth time, ie getting them there in about 0.44 of the time observed on earth, and exactly twice the time it would take light to appear to get there.

So, not only can you go as fast as you like, you can also travel as far in the future as you like. For example, to travel 100 years into the future, just get in a spaceship armed with 10000 times your own mass in matter-antimatter fuel, and then travel at Warp 100 for one year. When you reach your destination, one year will have passed for you, and 100 years (plus a little bit) will have passed on earth.

Of course, then you’ll have to get back to earth, so good luck with that.

The Buckaroo Banzai Principle

The Buckaroo Banzai Principle

No matter where you go — there you are.
— Buckaroo Banzai

The point of this exercise is that if you really understand what Einstein said, the idea should be that there is no absolute frame of reference. What this means is that even if you are travelling at 99.999 % the speed of light relative to the earth, as far as you know everything still looks and feels like Newton’s physics, where F = ma and you can always accelerate faster and faster. And not only that, but if you are heading for a specific location, the faster you go, the faster you will get there.

No Free Lunches

Now having said that, there are some consequences that the universe may unleash should you decide to try to go Warp 100. This is because even though the physics of your spaceship will be the same even at this insane speed, you are also surrounded by the gases in your local galaxy, as well as all of the light from stars that are visible to you. And even though from the earth much of this light is nice, low-energy visible spectrum, and even though that light will still be reaching you at the speed of light, it’s relative energy is radically different when you are plowing through that light at Warp 100. In fact, what you will be observing is a massive Doppler-shifting into the deep blue/ultraviolet of all light coming at you in the direction you are headed (and conversely, red-shifted looking back towards earth). Some of this light may be equivalent to the powerful cosmic rays that hit the earth, and which were generated by massive explosions or quasars just after the Big Bang. The energy in these photons may be enough to kill you all by themselves, especially at Warp 100. You may need a very large and thick radiation shield, along with all the extra energy to carry that shield along with you and your ship.

And so as we already should have known, there are no free lunches. At least it is nice to know that a faster-than-light lunch is available, should one choose to pay the price.

New Year’s Koan

Koan: a paradoxical question or story, used (in Zen Buddhism) as an aide to meditation and as a means to lead one to enlightenment.

The main problem with New Year’s resolutions and the reason they fail, I think, is that they are in the form of commandments. Humans are contrary by nature and any dictate — even one they have given to themselves — is doomed. The thing that motivates people is curiosity, and so in that spirit I offer up the following questions, upon which the reader (including myself) may ponder, and should any insight be gained, I am hopeful that it will lead one to a more fulfilling or meaningful life in the future. As with most Koan, I have no answer to these questions, and have no expectation for you to answer them — just to think about them.

New Year’s Koan for Atheists

Koan A1: The atheist Christopher Hitchens once said “What can be asserted without proof can be dismissed without proof.” Suppose a close and lifelong friend tells you one day “I have been in love with you for years.” This comes as a complete (though pleasant) surprise to you. Should you dismiss this assertion without proof? Alternatively, do you demand proof or accept the statement on its face? Does Hitchens’ principle not apply in this case? If so why?

Koan A2: There is to date no scientific evidence that Free Will exists. Does it make a difference to you in how you experience the world by assuming that you do or do not have Free Will? In other words, do you live your life “as if” Free Will has been shown to exist and that you possess it? Would it make a difference if you learned that Free Will does not exist but is some kind of illusion? Why?

Koan A3: The atheist Chris Arnade is a former physicist who worked for Wall Street before working with and photographing homeless addicts in South Bronx. He observed that in these squalid homeless places, often empty shells of buildings, bibles were always found and that this bible was all these hopeless people had to carry them through the next day or hour. Suppose your otherwise healthy spouse or child told you they’ve given up and wanted to kill themselves, and by their attitude and mood you are convinced they are sincere and would carry out their threat. As an atheist, what could you tell them that would give them some hope or reason to carry on with their lives ?

New Year’s Koan for Judeo-Christians

Koan B1: In Exodus 32:14, God changes his mind about punishing Moses’s people who had become corrupt, after Moses reminds God about the promises God had made to Abraham, Isaac and Israel. Let us leave aside the puzzle of how an omniscient god could change his mind. Do you have enough faith to talk back to God himself as Moses did, if you believe He has made a terrible mistake? If you were to talk back, what would you say?

Koan B2: If the Judeo-Christian belief is correct, then among those nonbelievers who have not been saved from damnation are Socrates, Buddha, Gandhi, Richard Feynman, Ralph Vaughan Williams, Carl Sagan, Bertrand Russel, Isaac Asimov, Albert Camus, Bela Bartok, David Hume, Bertolt Brecht, Heraclitus, Anton Chekov, Billy Joel, Joseph Conrad, Sergei Prokofiev, Eric Hofer, Camille Saint-Saens, George Santayana, Jean-Paul Sartre, Paul Dirac, Sigmund Freud, and Alfred Nobel … to name a few. On the other hand, the Emperor Theodosius, who executed children for playing with pagan dolls, and Charlemagne, who beheaded 4500 Saxons that refused to convert, were Christian and therefore saved. There is no question, this is simply something to ponder.

Koan B3: Do you believe that you have a soul, the essence of your spiritual self, and that it is eternal? If so, are the things you are doing in your life really preparing you for an infinite amount of time in which to spend your days? Remember, a billion years is a blink of an eye in the face of infinity. What will you do? And how will you keep from becoming bored beyond all measure? Knowing yourself, do you expect that a day will come in a quadrillion years in which you long for an end to endless wakefulness, in other words to Die? If so, how do you distinguish this place from what most would call Hell ? If you had a choice, would you prefer that your soul was only finite in time, say a thousand or million years ? Three score years and ten ?

New Year’s Koan For Everybody

Koan E1: If the race of man is to advance, either through evolution or divine intervention, in what way could man as a species be improved? Having answered that, is there any way in which you can begin to manifest any of that change in your own life ?

Happy New Year.

Our house, December 2013.

Grow Up!

I love a good novel, and the thing that makes a novel good is that it pulls you in, and is so well written that you are willing to suspend disbelief and swim in that world until the last page. And when it is over you wish there were more, because the story was so good.

I don’t buy most of the world’s religions these days because their stories aren’t believable, and I find myself putting down the book on page one. And after all, what is “faith” but another word for the willful suspension of disbelief ?

The world languishes from the lack of a religion for adults, who have had rich and varied life experience, a fully developed cerebral cortex and capacity for reason, complex emotion, and powers of direct observation of phenomena. A religion that has matured to the point that it admits that it does not have all the answers and is as flawed as we are, subject to revision pending new insights and findings as they come along. A story written by an unreliable narrator, who admits as much in the telling of the tale, but you don’t care because the story rings true.

The reason I don’t buy the stories told by the major religions of the world is that they sound like they were written by very small children, to whom their parents are perfect, omniscient, and will always take care of them to their dying days. What I don’t see in any of these stories is the humility of an adult, who has learned that their parents were flawed creatures at best, just like themselves, who didn’t know all the answers but tried their best, and at least has some hope that even if their own lives were screwed up, maybe the children of their creation will learn and do something good with this mangled beautiful mess of a world.

If for the moment, I buy the part of the usual story that we are created in the image of the universe’s creator, I would have to conclude that the creator was a fairly good mathematician and an artist, but like myself also mortal, and painfully limited in foresight about the consequences of one’s work, but hopeful that something good might come of all this after they are gone.

The first steps into adulthood begin when you realize that your parents are gone, and it is time for you to pick up the baton and do something yourself, with the realization that everyone else is in the same boat, and to have compassion for their own struggle with existence. If there ever was a creator, I am sure that they are long gone, but I’d like to say thanks for the good work, we will take it from here — as Ayn Rand would say — In the Name of the Best Within Us.

That is what I would call a religion for adults.

The world, with its undetermined future, is a vast blank canvas, and if there is any meaning in all of this, it reveals itself when you create something on that canvas that is beautiful.

Time to grow up.

There is No Such Word as Have

I had a “revelation” of sorts in recent years, which in retrospect many religions would (rightly)claim they thought of first: which is that much needless suffering and misery comes from the use of the meaningless yet toxic word “Have”. The word “Have” is a story, and almost pure fiction, as about the only thing you can truly be said to have is this moment in time, and the choice presented to you in that moment. All else in this saha world is like water, which slips through your fingers no matter how tightly you grasp.

To put things in perspective, wherever you see the word “Have” replace it with “Take Care Of” (or similar active verb), and the issue will become clear. Here is a short list of examples:

1. You do not Have a baby.  The reality is, for the next 18 years you will be Raising an Adult Human Being. You are required to study for years and get a license to be a psychiatrist or teacher or financial planner (skills needed for this task), and yet all you require to be presented with a high-maintenance creature such as a baby is puberty and a poor sense of future consequences.

2. You do not Have a million dollars (when you win the lottery). The reality is, you need to Take Care of Money for it to grow (or even stay the same). The sad thing is, most people do not know what Money is and how it works. If you stuff it in a mattress, inflation will make its magical power vanish. If you buy things with it, and those things do nothing useful, then they will break down and you will again have nothing. If you suddenly are given a million dollars, what you would need to Do is to study finance and accounting, and try to find investments (ie, business verntures people Do) that are productive and profitable.

3. (My recent example) You do not Have a house. The reality is, There are weeds to pull, Adobe bricks to repair, air filters to replace, pool water pH chemistry to maintain, mortgages to pay, and so on. If you do not Do these things and Take Care Of the House, it will break down, burn down, be condemned, and one way or another, it will go away. If you don’t think you would enjoy Taking Care of a House, then don’t buy it.

4. (This one really pisses me off) You do not Have a puppy. You are entrusted with what will soon be a much larger and  highly energetic border collie mix, a herder breed, who by her nature needs to be walked several times a day, given her shots, kept active and busy or she will eat and destroy your plants, shoes, blankets etc, and needs to be trained not to herd your other pets and kids. You need to be aware of all of this before you give your kids the cute little puppy, so that later you don’t renege on your promise and one day in frustration drive the dog out to a lonely stretch of highway in the southwest Utah desert, and abandon it on the road to be killed by cars or coyotes. If you do such a thing you also abandon your right to be called a Human(e) being. Only if the animal is very lucky, will she find a home occupied by people who take responsibility for the creatures entrusted to them, because they know there is no such word as Have.

5. You do not have Have a Life. You are Living.

Other examples are left as an exercise to the reader.

The Day Santa Wept

[My mother’s article, reprinted from The Argus]

The Day Santa Wept

By Lucille Mieher


It was early December, 1957, and I was climbing the steps to the second floor of the sprawling brick building. Looking down at my my bright red costume, I thought how fitting for a woman 8 1/2 months pregnant to play Santa Claus. No stuffing needed. A naturally round “jolly old elf.”

Stopping at a landing to catch my breath and adjust my wide black belt, my thoughts went back to September when this had its beginning. The young women’s club I had just joined had several pet projects, one of which was entertaining a ward at our local state mental hospital. Kentucky was having its typical Indian summer, I was suffering from the heat, and quite frankly secretly dreading this philanthropic chore.

“Let’s forgo the Bingo games today,” said the activities director, “and take the patients for a walk on the grounds. It’s cooler out there.”

As we gathered the people together, I noticed one young woman sitting on the floor, her back to the wall, her arms holding her knees tight against her chest, her head bowed.

“What about her? I inquired.

“Oh, that’s Evelyn,” the patients volunteered in unison. “She never does anything. Leave her alone.”

Ignoring their advice, I went over and touched her shoulder.

“Evelyn?”

Slowly she raised her head and I saw that she was probably my age or a few years older. Her eyes were expressionless until she spotted my maternity top and the obvious condition beneath it.

“You’re pregnant, aren’t you.” It was more a statement than a question.

“Yes, I am, and we’re all going for a walk out under the trees. Come with us.” I held out my hand to her.

We went for a walk

She hesitated, then put her hand in mine and slowly rose from her crouched position.

As we walked, she began to talk, mostly about children. She asked questions: “When is your baby due? Is this your first? Do you want a bory or a girl?”

I said, “The baby is due in late December, I already have a three-year-old boy,l and I’m kind of hoping for another boy.”

Then it was my turn to do the quizzing, “Do you have children?”

She became pensive for a moment, and then, in almost a whisper, said, “Yes, I ahve two beautiful little girls, only a year apart in age. I’m not sure how long it’s been since I saw them. I do remember waving to them as they boarded the school bus. We live in the country.” Evelyn and I continued to walk and talk.

After the outing and a cool drink of lemonade, our club returned our charges to their ward. As I waved good-bye to my new friend, she called out, “Good luck. I hope you have another boy.”

Back as Santa

Now here I was back again, but this time eager, rather than apprehensive. I practiced my “Ho Ho Ho” one more time, and entered the ward.

As I handed out presents and candy canes to the patients, I kept scanning the faces in search of the one most familiar to me.

“Where’s Evelyn?” I asked the nurse who was plaing Santa’s helper.
“She’s not here.”

“Oh, no.” I was almost afraid to ask the next questino. “What happened to her?”

Seeing my distraught face, the nurse continued, “Don’t worry, it’s good. In Septermber — I guess it was soon after your group was here — she began to respond to treatment. She asked to see her husband and children. They visited her, and she was allowed to spend Thanksgiving Day at home with them. That went so well, and she showed so much improvement, the doctors granted here what we call a ‘furlough’ to spend the month of Deember with her family. With luck, she may never have to come back here.”

Needless to say, Santa stood there crying in his beard. Tears of joy.

Two weeks later, my son was born.

Book Review: Dreams From My Father

 Dreams from My Father

by Barack Obama

The woman who inspected my bag at the security gate of the Playboy Jazz Festival noticed the copy of Dreams from My Father that I had packed along with the wine and cheese. “Have you read it?” I asked. “No,” she said, “I’m a Republican.”

If I had asked whether she had ever heard Dave Brubeck in concert, I thought, would she have said, “No, I’m Episcopalian?” I took my bag and went inside, sad, puzzled and troubled by ill omens of things to come. I thought of Danny Glover’s character in the movie Grand Canyon in one scene where he said, “Oh, man, things ain’t supposed to be this way.”

Something has gone terribly wrong lately in the way people talk to other people, and by “other people” I mean people who do not happen to come from the same tiny enclave, church, cable channel, blog, twitter-feed or political clan. I don’t know why. It does not seem to be limited to the right or left, the religious or secular. There is a visceral, almost toxic reaction people seem to now have about “Others” that shuts down all discussion, all willingness to see the world through others’ eyes or even acknowledge that others may have legitimate reasons for thinking the way they do.

This is my second review of the Barack Obama book, Dreams from My Father. The first one I threw away. Which is sad, because I thought it was a very interesting study of Obama’s story in light of Joseph Campbell’s universal Hero-myth, comparing Obama’s spiritual search for his father with the myths of Orpheus, Gilgamesh, King Arthur, and yes, even Star Wars and Harry Potter.

All very interesting, yet none of it, I’m afraid, convincing to a security guard at the Hollywood Bowl to whom the face on the cover is in itself a conversation stopper. She doesn’t know what the book is about, and she doesn’t want to know. I’ve tried to think of what I could have said to this person that would have made her even consider the possibility of opening up a copy of the book and reading a paragraph or two. I could have tried saying something like this:

“Politics aside, this is a great book. Not only may you be surprised by what you learn about the author and his story, but you might learn something about yourself as well. And who knows? Wonder of wonders, you may actually find that you are enjoying the process of reading a well-written and engaging book by a talented author.”

. . . . which would all be true, but even then I doubt that she would be convinced. One look at his face on the cover and there is simply nothing to discuss.

And so we arrive at a major theme of Obama’s book: How can people from different worlds and different tribes ever learn to communicate with each other? Unless we learn how to do that there is no hope.

Dreams from My Father was based on work originally commissioned in 1991 by book publishers seeking to capitalize on the notoriety of a young Obama, fresh out of law school, and the first black editor of the prestigious Harvard Law Review. He had intended the book to be a systematic analysis and discussion of race relations in America, but as he confesses in the introduction, “When I actually sat down and began to write, though, I found my mind pulled toward rockier shores.”

Rocky shores indeed. As the son of Barack Senior, a black African man and Stanley Ann, a white mid-western woman, Obama’s very DNA is of two worlds conventionally considered mutually exclusive and antagonistic. Whether it was with his black friends’ refrain “that’s just how white folks will do you” or when his white grandmother was scared by a panhandler – not because he was big, but because he was black – the same theme emerges: people with whom he identified talked with fear about scary people from a foreign world called them, and Obama realized that in both cases he was them. In Dreams from My Father, after learning of his grandmother’s fear, he wrote:

The earth shook under my feet, ready to crack open at any moment. I stopped, trying to steady myself, and knew for the first time that I was utterly alone.

Indeed, the book is prefaced with a biblical epigraph from I Chronicles 29:15: “For we are strangers before thee, and sojourners, as were all our fathers.” While the significance of the passage in referencing fathers is clear, the key I believe lies in the word sojourner, a stranger, passing through a place not his home.

The author has a talent for writing clean simple prose and allowing emotions to emerge in concrete external things. As the story opens, Obama has just received a long-distance phone call from a stranger who says that she is his Aunt Jane in Nairobi, Kenya with news that his long-absent father was killed in a car accident. He describes the moment like this:

That was all. The line cut off, and I sat down on the couch, smelling eggs burn in the kitchen, staring at cracks in the plaster, trying to measure my loss.

I love this passage. In one sentence the writer has marshaled all five senses into an elegant haiku about a single moment, the un-measurable despair dripping from the plaster cracks. The book is filled with such word-craft, making one eager to press on through the narrative for further rewards.

Though he initially planned to travel to Africa to attend the funeral, Obama decided at last not to go. In spite of this, the tragic event of his father’s death sets in motion the idea of a journey within Obama, a need to come to terms with the stories told about his father, and the stories he told himself, and a quest to find and reclaim his own identity and find a place called Home

In part one, Origins, he traces his childhood first in Indonesia and then Hawaii, growing up with his white mother and grandparents, attending college and ending with his memory of a brief and only visit from his father. He determines that he must somehow find him. In part two, Chicago, he embarks on an idealistic project working as a community organizer in the south side of Chicago. Things do not go well at first and the presence of machine politicians with agendas gums up the works. He eventually begins to have small victories, and in the process meets many people who later become significant in his life such as the controversial and outspoken Reverend Wright who coined the phrase “The Audacity of Hope.” In the third and final part, Kenya he travels to Africa and meets the Obama side of his extended family. He learns the long story of the Obama clan and the battles fought between his father and authoritarian grandfather.  culminating in a profound and gut-wrenching epiphany by the author at the stone-covered grave of the father he never knew.

The people who populate his story are complex and three dimensional human beings. You care about them even if they sometimes annoy you. Obama does not let you off the hook with easy stereotypes such as the “Caring Mother” or the “Corrupt Chicago Mayor.” People have reasons for what they do and often have doubts. Even in his self-portrayal, Obama relates his inner thought processes and dialectic struggles, and admits that he does not know all the answers — in a way that makes me understand now why he can sometimes be infuriating to both sides of the political spectrum. For Obama, as for all of us, the answers are never simple and sometimes our opponents really do have valid points.

It is not a spoiler to tell you that in the end Obama returns from his vision-quest and the transformation at his father’s grave with newly found wisdom. If our security guard at the jazz festival could read only one passage from this book that is the distillation of the contents, I could do no better than recommend to her (and you) the concluding words Barack Obama addresses to his father regarding the great and dangerous powers of the modern world and its only hope:

…That this power could be absorbed only alongside a faith born out of hardship, a faith that wasn’t new, that wasn’t black or white or Christian or Muslim but that pulsed in the heart of the first African village and the first Kansas homestead – a faith in other people.

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