Deliberately Difficult to Watch

difficult to watch

I’ve never tried this before: create a rotating polyhedral image which is difficult to watch, using disorienting effects, such as the rotation of the images of spirals on the rotating faces. The spiral is made of golden gnomons (obtuse triangles with a base:leg ratio which is the golden ratio). This image, alone and without comment, is shown in the previous post, and was made using Geometer’s Sketchpad and MS-Paint. In the preparation for this post, it was further altered, including the projection of it onto the faces of a great rhombicosidodecahedron, and creating this rotating .gif. This part of the process was performed using a program called Stella 4d: Polyhedron Navigator, available here. You be the judge, please: is it, in fact, difficult to watch? Did I accomplish my (admittedly rather odd) goal?

The Second of Dave Smith’s “Bowtie” Polyhedral Discoveries, and Related Polyhedra

Dave Smith discovered the polyhedron in the last post here, shown below, with the faces hidden, to reveal how the edges appear on the back side of the figure, as it rotates. (Other views of it may be found here.)

Smith's puzzle

So far, all of Smith’s “bowtie” polyhedral discoveries have been convex, and have had only two types of face: regular polygons, plus isosceles trapezoids with three equal edge lengths — a length which is in the golden ratio with that of the fourth side, which is the shorter base.

Smiths golden trapezoid

He also found another solid: the second of Smith’s polyhedral discoveries in the class of bowtie symmetrohedra. In it, each of the four pentagonal faces of the original discovery is augmented by a pentagonal pyramid which uses equilateral triangles as its lateral faces. Here is Smith’s original model of this figure, in which the trapezoids are invisible. (My guess is that these first models, pictures of which Dave e-mailed to me, were built with Polydrons, or perhaps Jovotoys.)

24-a-gon_HDR

With Stella 4d (available here), the program I use to make all the rotating geometrical pictures on this blog, I was able to create a version (by modifying the one created by via collaboration between five people, as described in the last post) of this interesting icositetrahedron which shows all four trapezoidal faces, as well as the twenty triangles.

Smith's Icositetrahedron

Here is another view: trapezoids rendered invisible again, and triangles in “rainbow color” mode.

Smith's Icositetrahedron H

It is difficult to find linkages between the tetrapentagonal octahedron Smith found, and other named polyhedra (meaning  I haven’t yet figured out how), but this is not the case with this interesting icositetrahedron Smith found. With some direct, Stella-aided polyhedron-manipulation, and a bit of research, I was able to find one of the Johnson solids which is isomorphic to Smith’s icositetrahedral discovery. In this figure (J90, the disphenocingulum), the trapezoids of this icositetrahedron are replaced by squares. In the pyramids, the triangles do retain regularity, but, to do so, the pentagonal base of each pyramid is forced to become noncoplanar. This can be difficult to see, however, for the now-skewed bases of these four pyramids are hidden inside the figure.

J90 disphenocingulum

Both of these solids Smith found, so far (I am confident that more await discovery, by him or by others) are also golden polyhedra, in the sense that they have two edge lengths, and these edge lengths are in the golden ratio. The first such polyhedron I found was the golden icosahedron, but there are many more — for example, there is more than one way to distort the edge lengths of a tetrahedron to make golden tetrahedra.

To my knowledge, no ones knows how many golden polyhedra exist, for they have not been enumerated, nor has it even been proven, nor disproven, that their number is finite. At this point, we simply do not know . . . and that is a good way to define areas in mathematics in which new work remains to be done. A related definition is one for a mathematician: a creature who cannot resist a good puzzle.

A Euclidean Construction of a Golden Rectangle in Which All Circles Used Have Radius One or Two

There is more than one way to construct a golden rectangle using the Euclidean rules, but all the ones I have seen before use circles with irrational radii. This construction, which I believe to be new, does not use that shortcut, which helps explain its length. The cost of avoiding circles of irrational radius is decreased efficiency, as measured by the number of steps required for the entire construction.

In the diagram below, the distance between points A and B is set at one. All of the green circles have this radius, while the magenta circles have a radius exactly twice as long.

UCGRC

To make following the construction from the diagram above easier, I named the points in alphabetical order, as they appear, as the construction proceeds. The yellow rectangle is the resulting golden rectangle. The blue right triangle is what I used to get a segment with a length equal to the square root of five, which is a necessary step, given that this irrational number is part of the numerical definition of the exact value of the golden ratio (one-half of the sum of one and the square root of five). In order to make the hypotenuse have a length equal to the square root of five, by the Pythagorean Theorem, the two legs of this triangle have lengths of one and two.

The Golden Ratio: Working from a Definition to Find a Value

800px-Golden_ratio_line.svg

I found the image above through the Wikipedia article on the golden ratio. After using what appears above to define the golden ratio, the article then reveals its exact and approximate values. Later, the writers of the article do show the calculations involved in doing this, but they seem unnecessarily complicated. I’m going to try to simplify the process here, and might later edit/simplify this Wikipedia article to make it more understandable.

So, first, “a + b is to a as a is to b” need to be written as a fraction, which is easy enough: (a + b)/a = a/b. The value of this fraction, a/b, is, by definition, φ, the golden ratio. As an equation, this can be written a/b = φ.

Next, apply cross-multiplication to (a + b)/a = a/b, and it becomes (a + b)(b) = (a)(a), which simplifies to ab + b² = a².

Also, since a/b = φ, this means that a = φb (via the multiplication property). Next,  ab + b² = a² is rewritten, with φb substituted for each a. The result of this substitution is (φb)(b) + b² = (φb)², which then becomes φb² + b² = φ²b². To simplify this, b² may be cancelled (via the division property), producing φ + 1 = φ². This may then be rearranged (via the subtraction and symmetric properties) to φ² – φ – 1 = 0. Two values of  φ can then be found via the quadratic formula, and they are {1 ± sqrt[1 – (4)(1)(-1)]}/2 = [1 ± sqrt(5)]/2. Use “+,” and calculate a decimal approximation for this irrational number, and you get ~1.618, which is the golden ratio. Use “-” instead, and you get a negative number (approximately -0.618), which can be rejected on the grounds that a ratio of two lengths must be positive, since all lengths, themselves, are positive.

Also, I’m changing my mind regarding changing Wikipedia, on this subject. The two versions of the calculation (the one now on Wikipedia, and mine) don’t match, but both are mathematically valid — and, while my version makes more intuitive sense to me, that doesn’t mean it would make more sense to others, and Wikipedia isn’t there for me alone. Until I actually wrote the calculation out, I thought my version would be simpler, but I cannot claim that now.

[Later addition: see the first comment below for a way, suggested by a reader of this blog, to simplify the calculation, as I wrote it above. I’m not going to take credit for his improvement, of course — that would violate mathematical etiquette!]

A Regular Decagon, Decomposed into Golden Triangles and Golden Gnomons

decagon

The golden triangles, in yellow, are acute isosceles triangles with a leg:base ratio which is the golden ratio. Golden gnomons, shown in orange, are related, for they are obtuse isosceles triangles where the golden ratio shows up as the base:leg ratio, which is the reciprocal of the manifestation of the golden ratio which appears in the yellow triangles.

A Golden Tessellation

golden tiling

This tessellation can be viewed in at least two ways: it can be seen as being composed of overlapping octagons which are equilateral, but not equiangular — or it can be viewed as a periodically-repeating pattern of golden gnomons, as well as golden triangles of two different sizes. Both golden triangles and golden gnomons are isosceles triangles with sides in the golden ratio, but golden triangles are acute, while golden gnomons are obtuse.