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
Just a quick note here: for those of you who have not heard of
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
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
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
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
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,
So in the picture on the right, the horizontal plane represents space at time
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
It turns out that we can even be very specific about how
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.
And yes, this is how I spend my holiday vacations. Having Fun ! Happy new year !