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Math humor
So I'm reading a paper on Sierpinski Pedal Triangles (a generalization on the famous Sierpinski Triangle fractal) and I come across this humorous bit in a proof of one of the theorems in the paper:

This completes the proof of Theorem B.

Oh it does, does it?

If you take an acute triangle -- one with all 3 angles less than aright angle -- there are several ways to inscribe a smaller triangle in it. If you connect the mid-point of each edge, you divide the triangle into 4 smaller triangles, each shaped just like the original. We can call this inscribed triangle the "midpoint triangle".

The Sierpinski Triangle (ST) is a famous self-similar fractal made effectively by taking the area of a triangle, discarding the area of the midpoint triangle so you have three triangles remaining with a hole in the middle, and then using each of those to make a Sierpinski Triangle (so you start with 1 triangle, then have 3 each 1/4 in area, then have 9 each 1/16 in area, then 27 each 1/64 in area, etc, to infinity).

There is another way to divide the original triangle that has some nice properties.

Traditionally, The "altitude" of a vertex of a triangle is a line segment drawn from the point to the opposite side such that the altitude is perpendicular to that side. This line is the shortest line between the vertex and the opposite side (often called the "base") so the altitude in a way measures the height of the triangle (especially in the commonly known "base times height divided by 2" formula for measuring a triangle's area. The point where the altitude meets the base is known as the "foot" of the altitude, since it's what the altitude "stands" on. Naturally, in a triangle there are three points, thus three altitudes and three feet.

If you inscribe a triangle using the three feet of a larger triangle, you get the "pedal triangle" of the original triangle. In general, the pedal triangle is not the same shape as the original triangle, but the three other triangles formed by the inscription are in fact the same shape, although they may be mirror-images. They are not the same size as each other (generally).

A "Sierpinski Pedal Triangle" (SPT) is formed by the same process as a Sierpinski Triangle, only using the pedal triangle instead of the midpoint triangle: Take a triangle, and discard the pedal triangle, yielding three smaller triangles similar to the original. Use each of those smaller triangles to make a Sierpinski Pedal Triangle.

Images of Sierpinski Triangles all look pretty much the same, although stretched and skewed. All Sierpinski Triangles are "Affine Equivalent", meaning that by a combination of translation, rotation, stretching and skewing you can transform any Sierpinski Triangle into another.

SPTs, on the other hand, can and do look different from each other. They look like an ST that's been warped a bit, and are not (generally) affine equivalent to STs or to each other. You can see images of a few in the PDF file linked above.


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