Tessellation of Bowtie Hexagons
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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.)
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.
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.)
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.
Here is another view: trapezoids rendered invisible again, and triangles in “rainbow color” mode.
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.
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.
The First of Dave Smith’s “Bowtie” Polyhedral Discoveries: An Example of Mathematical Collaboration
Recently, a reader of this blog contacted me about a polyhedron he wished to model. His name is Dave Smith, and he had already done much of the work involved, but needed help finishing off his project. Here’s the picture he e-mailed me.
The visible faces are regular pentagons — four of them. The invisible faces are isosceles trapezoids, in two “bowtie” pairs which share their shortest edges with those of their reflections. I e-mailed Smith, and told him the truth: I didn’t have a clue how to make this in Stella 4d, the program I use to make the rotating polyhedra on this blog (including the one below). I also told him I wasn’t giving up — merely enlisting help with his puzzle.
And, with that, I went to Facebook, posting the image above, along with an explanation, and request for help finishing it. This may not be what most people think of when they consider Facebook, but I have deliberately sought out experts there in many fields, including geometry, to make the social-networking site useful in unusual ways, such as getting help with geometrical puzzles I can’t solve alone. Three geometricians with skills which exceed mine (Wendy Krieger, Tom Ruen, and Robert Webb, who wrote Stella 4d) began discussing the figure. One of them, Tom Ruen, sent me .stel files (That’s what Stella 4d uses) for multiple figures, getting closer each time. With the last such virtual model Tom sent me, I was able to “tweak” it to get the pentagons regular.
This eight-faced figure has two edge lengths, the shorter appearing only twice, as the shared, shorter base within each “bowtie” pair of isosceles trapezoids — and these two edge lengths are in the golden ratio. A type of octahedron, it also has an interesting form of symmetry — it reminds me of pyritohedral symmetry, but is not; the features seen in pyritohedral symmetry in relation to the x-, y-, and z-axes of coordinate space only show up here in relation to two of these three axes. This symmetry-form is called dihedral symmetry.
And it only took five people to figure all of this out!
A Radial Tessellation of Regular Decagons and Bowtie Hexagons
This tiling-pattern could be continued indefinitely, while still maintaining its five-fold radial symmetry, giving it the overall appearance of a pentagon.
Two Symmetrohedra Featuring Regular Pentadecagons
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I’ve posted “bowtie” symmetrohedra on this blog, before, which I thought I had discovered before anyone else — only to find, later, that other researchers had found the exact same polyhedra first. Those posts have now been edited to include credit to the original discoverers. With polyhedra, finding something interesting, for the first time ever, is extremely difficult. This time, though, I think I have succeeded — by starting with the idea of using regular pentadecagons as faces.
Software credit: Stella 4d was the tool I used to create this virtual model. You can try a free trial download of this program here: http://www.software3d.com/Stella.php.
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Update: once again, I have been beaten to the punch! A bit of googling revealed that Craig Kaplan and George Hart found this particular symmetrohedron before I did, and you can see it among the many diagrams in this paper: http://archive.bridgesmathart.org/2001/bridges2001-21.pdf.
You’ll also find, in that same paper, a version of this second pentadecagon-based symmetrohedron:
There is a minor difference, though, between the Kaplan-Hart version of this second symmetrohedron, and mine, and it involves the thirty blue faces. I adjusted the distance between the pentadecagons and the polyhedron’s center, repeatedly, until I got these blue faces very close to being perfect squares. They’re actually rectangles, but just barely; the difference in length between the longer and shorter edges of these near-squares is less than 1%. I have verified that, with more work, it would be possible to make these blue faces into true squares, while also keeping the pentadecagons and triangles regular. I may actually do this, someday, but not today. Simply constructing the two symmetrohedra shown in this post took at least two hours, and, right now, I’m simply too tired to continue!
A Cube-Based “Bowtie” Symmetrohedron Featuring Six Regular Hexadecagons, Eight Equilateral Triangles, and Two Dozen Each of Two Types of Icosceles Trapezoid
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The two types of trapezoid are shown in blue and green. There are twenty-four blue ones (in eights set of three, surrounding each triangle) and twenty-four green ones (in twelve sets of two, with each set in “bowtie” formation).
This symmetrohedron follows logically from one that was already known, and pictured at http://www.cgl.uwaterloo.ca/~csk/projects/symmetrohedra/, with the name “bowtie cube.” Here’s a rotating version of it.
(Images created with Stella 4d — software you can try yourself at http://www.software3d.com/Stella.php.)
A Bowtie Symmetrohedron Featuring Twelve Decagons and Twenty Equilateral Triangles
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Created using software you can try at http://www.software3d.com/stella.php.
Later edit: I found this same polyhedron on another website, one that has been online longer than my blog, so I now for, for certain, that this was not an original discovery of my own. At http://www.cgl.uwaterloo.ca/~csk/projects/symmetrohedra/, it is named the “alternate bowtie dodecahedron” by Craig Kaplan and George W. Hart.
Tessellation of Regular Dodecagons and Nonagons, Together with “Bowtie” Hexagons
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A “Bowtie” Polyhedron Featuring Regular Enneagons and Octagons
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So far as I know, no one knows how many otherwise-regular convex “bowtie” polyhedra exist — that is, convex polyhedra whose only faces are regular polygons, and pairs of isosceles trapezoids in “bowtie” formation. With the aid of software called Stella 4d, which you can find at http://www.software3d.com/Stella.php, I do believe I’ve found another one which hasn’t been seen before.
To make it, I started with what is probably the most well-known near-miss to the Johnson Solids, this polyhedron featuring enneagons (nine-sided polygons; also called “nonagons”):
I then augmented each enneagonal face with regular antiprisms, took the convex hull of the result, and then used Stella’s “try to make faces regular” function — and it worked, making the octagons regular, as well as the enneagons.
Update: It turns out that this polyhedron has been seen before. It’s at http://www.cgl.uwaterloo.ca/~csk/projects/symmetrohedra/ — and there are even more at http://www.cgl.uwaterloo.ca/~csk/papers/kaplan_hart_bridges2001.pdf. These include several more “bowtie” polyhedra found among what those researchers, Craig S. Kaplan and George W. Hart, call “symmetrohedra.” They call this particular polyhedron a “bowtie octahedron.”
A “Bowtie” Expansion of the Truncated Dodecahedron
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This polyhedron has the twelve regular decagons and twenty regular triangles of the truncated dodecahedron, but they are moved outwards from the center, and rotated slightly, creating gaps. These gaps are then filled with thirty pairs of isosceles trapezoids in “bowtie” formation. That gives this polyhedron 92 faces in all.
Software credit: see http://www.software3d.com/stella.php
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