Four Polyhedra Featuring Heptagons

Heptagons only appear infrequently in interesting polyhedra. I recently found a few that I like.

Icosidodeca 2nd tetstell with heptagons and triangles is pyritohedral.gif

To form the first of these solids, shown above, I started with the icosidodecahedron, dropped the symmetry of the model from icosahedral to tetrahedral, and then stellated it twice using Stella 4d (available here). To obtain the model shown below, which also features heptagons and triangles, I stellated it once more. Both of these polyhedra have pyritohedral symmetry.

Icosidodeca 3rd tetstell with heptagons and triangles is pyritohedral.gif

To form the next model shown, I began with an rhombicosidodecahedron, set it to tetrahedral symmetry, and stellated it eight times. This produces a chiral solid with tetrahedral symmetry.

Stellated Rhombicosidodeca 8th tetstell features 8 heptagons tet symmetry.gif

For the last of these four polyhedra featuring heptagons, I began with the snub dodecahedron, dropped the symmetry of the model down from icosahedral to tetrahedral, and then stellated it sixty-one times. The resulting solid is chiral, with tetrahedral symmetry.

Snub Dodeca 61st tetstell.gif

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A Pyritohedral, Stellated Polyhedron, and Its Convex Hull

To make this polyhedron using Stella 4d (available here), I began with the dodecahedron, dropped the symmetry of the model from icosahedral to tetrahedral, and then stellated it thirteen times. 

Dodeca 13th tetstell.gif

This stellated polyhedron has pyritohedral symmetry, but this is easier to see in its convex hull:

Convex hull of the dodecahedron's 13th tetstell.gif

The eight blue triangles in this convex hull are equilateral, while the twelve yellow ones are golden isosceles triangles.

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A Chiral Polyhedron Made of Kites and Triangles, Along with Its Dual, Made of Triangles and Isosceles Trapezoids

To make this polyhedron using Stella 4d (available here), one starts with the icosahedron, drops the symmetry of the model down from icosahedral to tetrahedral, and then stellates it once. The result is a chiral solid featuring four triangular faces and twelve kites:

chiral polyhedron made of triangles and kites -- found while exploring tetstells of the icosahedron.gif

The dual of this polyhedron, which is also chiral, has four triangular faces, and twelve faces which are isosceles trapezoids. It is a type of faceted dodecahedron — a partial faceting, meaning it is made without using all of the dodecahedron’s vertices.

Faceted Dodeca.gif

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Compelling Reading, I’m Sure, from Facebook

compelling reading

So, did anyone out there actually read this? If so, how is the plot, and the character development?

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The Liebster Award for Blogging

I’d like to thank Richard, at https://photosociology.wordpress.com/, for nominating me for the Liebster Award for blogging. It’s an honor to simply have someone choose to follow my blog, and I feel grateful every time I get a new follower, but this takes that feeling of being honored to a whole new level.

liebster1

These are the guidelines for the 2018 Liebster award:

• Thank the person who nominated you
• Display the award on your post
• Write a small post about what makes you passionate about blogging
• Provide 10 random facts about yourself
• Answer the questions given to you
• Nominate 5-11 other blogs for this award
• Ask them creative and unique questions of your own
• List the rules and inform your nominees of the award

What makes me passionate about blogging? Well, for as long as I can remember, I have been passionate about mathematics, to the point of obsession. Blogging gives me a way to record and share the results of that obsession. My blog isn’t 100% math, but mathematics-related posts outnumber everything else here by a wide margin, and it is my love of mathematics that keeps this blog going.

On to the ten random facts about myself . . .

  1. I have Aspergers Syndrome, now officially known in the USA as high-functioning autism. I didn’t discover this until I was already in my 40s (I’m now 50), though, for which I am grateful. I see being an “Aspie” as a difference, not a disease, nor a disability.
  2. I’m married to a wonderful woman; we celebrate our 4th wedding anniversary soon.
  3. I’m a high school teacher. Next year will be my 24th year in the classroom. I mostly teach mathematics and the “mathy” sciences. My wife is a teacher as well; she teaches mathematics.
  4. Strangely enough, both of my college degrees are in history. This generally puzzles people, but it’s easy enough to explain: I chose to major in something I didn’t yet know much about, and about which I was (and still am) curious. My experiences in elementary, junior high, and high school math classes were abysmal, and I didn’t care to continue that experience.
  5. I’m not religious. The label I prefer is not “atheist” nor “agnostic,” though, but simply “skeptic.” This reflects the fact that I have two primary methods for determining what I consider to be true: mathematical proof, and the scientific method. Skepticism is essential for both.
  6. I see my brain as an organic computer, and frequently work on re-writing my own software, usually while asleep. This is something I’ve blogged about before, as are most of the things in this list.
  7. I started blogging on Tumblr, and came to WordPress a few years later, in 2012, to escape what I call Tumblr’s “reblogging-virus.”
  8. My political orientation has changed over the years, and is now best captured by the term “anti-Trumpism.” I’ve also been known to call myself a “neo-Jeffersonian.”
  9. I’m LGBTQ-friendly.
  10. I’ve seen the fantastic band Murder By Death seven times. Here is a sample of their music, from one of their older albums. You can find much more about them at http://www.murderbydeath.com.

Now I need to answer the questions which Richard has posed for me. I have a hunch my “Aspieness” will come out in some of these answers.

1. How straight is straight?

“For any two points, there is exactly one line which contains them.” This is a fundamental postulate of Euclidean geometry. Straightness is a characteristic of such lines.

2. What would you think I was referring to if I told you to ‘put it down’?

The contents of my hand(s), of course. If I wasn’t holding anything, I’d simply be confused, and would ask for clarification.

3. Why are swans graceful?

Swans have the characteristics they have because they evolved that way. It is human beings who have chosen to label some of those characteristics as “gracefulness.”

4. Would you be a superhero or a sidekick, and what would your name be?

I would do neither, for I have at times suffered from delusions that I had superpowers. I don’t want my mind to go there again. One example of this was a belief, years ago at a time of ridiculously high stress, that my emotional state could control the weather. If I start thinking I have superpowers again, I’ll immediately take the medication prescribed by my psychiatrist for just such an eventuality. 

5. If you could remove one letter from the English alphabet, what would it be, and what consequences do you see coming from it?

I suppose I would choose the letter “c,” for the soft “c” can be replaced by the letter “s,” and the hard “c” by the letter “k.” I’m not sure what we’d do for the “ch” sound, though.

6. What was the last thing you lost and never found? What do you imagine has happened to it?

That’s my Social Security card, which I need to get replaced soon. I don’t have a clue what happened to it.

7. What significance does the number seven have to you? What memories do you associate with it?

I’ve blogged about the significance of the number seven, so I refer you to that post for the answer to this question. The only memory of the number seven I recall is when a friend of mine named Tony explained to me the ideas which later inspired that blog-post.

8. Young and completely broke or old and disgustingly rich?

Neither, by the standards of where I live (the USA). We’re middle-class. We live comfortably, but not extravagantly.

9. If a giant squirrel had commandeered your mode of transportation, whether car, moped, bike etc., and seemed to know how to make it work, what would you do to stop him?

I would assume this was a hallucination, and I would immediately take the medication I mentioned when I answered question #4, above.

10. If you had your own coat of arms, what would I expect to find on them to describe you/ your family?

Some of my ancestors were Scottish, and their clan already has a coat of arms, so I’d simply use it.

Clan Keith

Next, here are my nominees for this award. These are all blogs I find interesting. I also deliberately chose blogs which are radically different from my own.

Robert Vella’s The Secular Jurist

The hilarious blog Seinfeld Law

da-AL’s Happiness Between Tails

The blog https://bananartista.net/

The blog https://swo8.wordpress.com/

Now I need some questions for these fine bloggers to answer:

  1. Do you see the current occupant of the White House as a problem? If so, what, if anything, are you doing about it?
  2. How strong a role does mathematics play in your life?
  3. Which of the sciences do you find most interesting, and why?
  4. Of all the posts on your blog, which one do you think is your best work?
  5. What food(s), if any, could you absolutely not give up for the rest of your life, even for $100,000?
  6. What do you think of astrology?

That’s it! Thanks again to Richard for nominating me; I’m glad I took the time to write this acceptance-post. Also, congratulations to the five new nominees!

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A Second Type of Double Icosahedron, and Related Polyhedra

After seeing my post about what I called the “double icosahedron,” which is two complete icosahedra joined at one common triangular face, my friend Tom Ruen brought my attention to a similar figure he likes. This second type of double icosahedron is made of two icosahedra which meet at an internal pentagon, rather than a triangular face. Tom jokingly referred to this figure as “a double patty pentagonal antiprism in a pentagonal pyramid bun.”

Augmented Gyroelongated Penta Pyramid one color

It wasn’t hard to make this figure using Stella 4d, the program I use for polyhedral manipulation and image-creation (you can try it for free here), but I didn’t make it out of icosahedra. It was easier to make this figure from gyroelongated pentagonal pyramids, or “J11s” for short. This polyhedron is one of the 92 Johnson solids.

J11.gif

To make the polyhedron Tom had brought to my attention, I simply augmented one J11 with another J11, joining them at their pentagonal faces. Curious about what the dual of this solid would look like, I generated it with Stella.

Augmented Gyroelongated Penta Pyramid dual.gif

The dual of the double J11 appears to be a modification of a dodecahedron, which is no surprise, for the dodecahedron is the dual of the icosahedron.

I next explored the stellation-series of the double J11, and found several attractive polyhedra there. This one is the double J11’s 4th stellation.

Augmented Gyroelongated Penta Pyramid 4th stellation.gif

The next polyhedron shown is the double J11’s 16th stellation.

Augmented Gyroelongated Penta Pyramid 16th stellation.gif

Here is the 30th stellation:

Augmented Gyroelongated Penta Pyramid 30th stellation.gif

I also liked the 43rd:

Augmented Gyroelongated Penta Pyramid 43rd stellation.gif

The next one shown is the double J11’s 55th stellation.

Augmented Gyroelongated Penta Pyramid 55th stellation.gif

Finally, the 56th stellation is shown below. These stellations, as well as the double J11 itself, and its dual, all have five-fold dihedral symmetry.

Augmented Gyroelongated Penta Pyramid 56th stellation

Having “mined” the double J11’s stellation-series for interesting polyhedra, I next turned to zonohedrification of this solid. The next image shows the zonohedron based on the double J11’s faces. It has many rhombic faces in two “hemispheres,” separated by a belt of octagonal zonogons. This zonohedron, as well as the others which follow, all have ten-fold dihedral symmetry.

Zonohedrified Augmented Gyroelongated Penta Pyramid f.gif

Zonohedrification based on vertices produced this result:

Zonohedrified Augmented Gyroelongated Penta Pyramid v.gif

The next zonohedron shown was formed based on the edges of the double J11.

Zonohedrified Augmented Gyroelongated Penta Pyramid e.gif

Next, I tried zonohedrification based on vertices and edges, both.

Zonohedrified Augmented Gyroelongated Penta Pyramid v e.gif

Next, vertices and faces:

Zonohedrified Augmented Gyroelongated Penta Pyramid v f.gif

The next zonohedrification-combination I tried was to add zones based on the double J11’s edges and faces.

Zonohedrified Augmented Gyroelongated Penta Pyramid e f.gif

Finally, I ended this exploration of the double J11’s “family” by adding zones to build a zonohedron based on all three of these polyhedron characteristics: vertices, edges, and faces.

Zonohedrified Augmented Gyroelongated Penta Pyramid v e f.gif

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Augmenting, and Then Reaugmenting, the Icosahedron, with Icosahedra

A reader of this blog, in a comment on the last post here, asked what would happen if each face of an icosahedron were augmented by another icosahedron. I was also asked what the convex hull of such an icosahedron-cluster would be. Here are pictures which answer both questions, in order.

Augmented Icosa with more icosas.gif

Convex hull of icosa augmented with icosas.gif

While the icosahedron augmented by twenty icosahedron forms an unusual non-convex shape, its convex hull is simply a slightly “stretched” version of the truncated dodecahedron, one of the Archimedean solids.

The reader who asked these questions did not ask what would happen if the icosahedron-cluster above were to be augmented, on every face, by yet more icosahedra. However, I got curious about this, myself, and created the answer: the following cluster of even-more numerous icosahedra. This could be called, I suppose, the “reaugmented” icosahedron.

Augmented Icosa with more icosas and then yet more icosas.gif

Finally, here is the convex hull of this even-larger cluster. No one asked for it; I simply got curious.

Convex hull of the reaugmented icosahedral cluster

To accomplish the polyhedron-manipulation and image-creation for this post, I used a program called Stella 4d: Polyhedron Navigator, which is available at http://www.software3d.com/Stella.php. A free trial download is available there, so you can try the software before deciding whether or not to purchase it. 

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