From Rabbits to Roses

This is the promised continuation of our mystery series "The Rose and the Nautilus". Last we met, we saw the connection of Fibonacci's down-to-earth sequence of "rabbit numbers"

with the elusive and legendary "Divine Proportion" --- incarnated by rectangles with a strange property: when you cut a square away from one of them, you are left with a rectangle of the same shape --- so you can continue cutting off squares indefinitely. So far, we don't know for sure that such rectangles exist -- they may be a pipe dream or a pie in the sky. All we have our hands on, so far, are number sequences like the one above, where two succesive numbers (high enough in the sequence) give us an approximately golden rectangle.

Today we'll make the journey from the golden rectangle to the golden triangle the major building block of the pentagram (the five-cornered star on US military vehicles), and hence the Pentagon (home of the US military brass). How fitting that the emblem of these admirable institutions shares its basic symmetry with the rose and many other flowers, from buttercups to petunias!

On the right you see a rose -- the underlying pentagram (shown in light blue) is made up of golden triangles, each of which is obtained from a golden rectangle by collapsing one of the shorter sides to a point. And if we do not have a perfect golden triangle handy, we'll take an approximate one made of rabbit numbers.

If you were given the yellowish triangle standing in the middle of Figure 1, you would not have much trouble filling in the rest of the diagram, would you? Alright -- the proportions of this triangle are determined by a=144 (base) and a+b=233 (side) units of length -- two successive rabbit numbers. And the pentagram connection? Oh, if we knew that the base angle of the big yellowish triangle is equal to twice its vertex angle, the latter would be 1/5 of 180 degrees -- see? Now , a/b is very nearly equal to (a+b)/a -- which means that the lower pale yellow triangle is well-nigh isoscles, and the distance from the lower left red dot to the blue dot is nearly equal to a. Hence the upper deep yellow triangle is well-nigh isosceles, and its exterior angle at the blue dot is roughly twice the vertex angle -- Q.E.D.

Do I hear you grumbling? You are not happy with all this uncontrolled fuzziness? You'd like crisp, clean equality instead of all those "well-nighs"? Well, last year we tried to work with the socalled "real" numbers -- and you hated them when you were asked to produce crisp, clean proofs. And then somebody said that it all came down to Cauchy sequences and/or Dedekind cuts plus continuity, and when we tried that approach you all freaked out. Let's face it: if you want numerical proportions, you have the choice between honest imperfection (e.g., rabbits) or dishonest perfection (e.g., root five). This is not rocket-science: Eudoxos realized it way back around 400 B.C. So let's get back to geometry, shall we? I admit that we have not yet constructed a perfect golden rectangle, but I promise you it isn't hard. For today, let us suppose it done and continue from there.

Our task is to repeat the argument pertaining to the pentagram connection (Figure 1) without any "nearly", "roughly" or "well nigh". Just go over it again, and you'll see that everything will be perfectly exact, as long as the pale yellow triangle in the lower right of Figure 1 is truly isosceles.

Into Geometry

In Figure 2, we are looking at a golden rectangle with the square at the bottom in two shades of yellow. The residual rectangle at the top is gray, and a copy of it is shown (light yellow) inside the square. As this gray thing as well as its copy is similar to the original big rectangle, the two red diagonals are parallel. Imagine them as rubber bands, and the dot in the lower left corner as a hinge. Of course, we are not dealing with rubber bands and hinges, but with geometric entitities in our minds. Yet this metaphor will avoid tedious descriptions, and also suggests that we speak of the brown vertical and horizontal sides as bars attached to that hinge.

In Figure 3, you now see how we make a big isosceles triangle by simply swinging the vertical bar through a small angle. The rubber bands will remain parallel, that is important, because it means that the smaller triangle in the lower left has the same base angles as the big one, do you agree? You are not quite convinced?

How perceptive of you -- and right you are: mechanics is not geometry. I promise to come back to those rubber bands. But first let us see how to get a pentagram from the triangle of Figure 3, supposing that the parallelitity holds up. The two-tone yellowish triangle in Figure 1 is, of course, our double triangle from Figure 3, with the smaller copy laid on its side. Does everything fit as it should?

First the angles: since big and small have the same base angles, small fits snugly into the right lower corner of big. Then the sides: the one labelled "a" in Figure 3 was the side of the square in Figure 2, hence fits exactly along the base. So it works! But remember: now that we have moved into geometry, "a" and "b" are line segments not numbers.

Desargues.

Here he is, on the left, staring at his theorem (its "affine" version) shown on the right. What does it look like to you? A leaning tower with two platforms in it? Good! What it says is this: if two triangles ABC and A'B'C' are lined up in so that the lines AA', BB', CC' meet in a single point, then if AB is parallel to A'B' and BC parallel to B'C', it follows that AC is also parallel to A'C'. Obvious? It almost would be if we were looking at a spatial set-up, as you suggested. Then the hypotheses would quickly imply that the entire two "platforms" are parallel. But this is meant as a flat, planar diagram, and, as I said, The theorem is impossible to prove within purely planar geometry.

We have gone way over-time -- let's prove Girard's theorem at our next meeting. You will see that the proof is clever but not hard to follow. We pretend that his diagram is the planar projection of a spatial gizmo, then transfer the hypotheses to the latter, draw our conclusion up there in space and finally project down again. For today let me just show you how it helps to keep our "rubber bands" parallel. Figure 4 shows the bare bones of Figures 2 and 3 superimposed. Do you see the rubber bands? In their original position (attached to the vertical bar) they are parallel. In Figure 5, we added two short red lines -- which are also parallel, being the bases of two isosceles triangles with the same vertex angle. Hence the rubber bands remain parallel in their new position (attached to the slanted bar). Et voilà !