geometry and paper · Solids

Six Weeks, Seven Shapes

Hebesphenomegacoron, J89

Some of my mathart friends suggested the #GeoInkoctober, #GeoInktober2021hastags as a way to be part of the recent October hashtag tradition. Having just had the gyrobifastigium on my mind, I looked to spend some time with the group of shapes called the Johnson Solids. There are 92 of these shapes, categorized by various shared attributes. At the end of the Wikipedia list there are 7 in the category of “Other.” I suppose that it’s predictable that this is the category I chose.

hebesphenomegacorona  stackedd
Hebesphenomegacorona, stacked

Didn’t quite know where messing around with these shapes would lead. Spent more than a week fussing with the hebesphenomegacorona, aka Johnson Solid 89, or J89. Used the same kind of faux (Photoshop) watercolor washes I’d recently used for other solids. Scaled the shape. Did lots of playing around with its template, even made it into a lamp.

There was something quite satisfying about working on this shape just enough to have something to post on twitter each day.

Triangular hebesphenorotunda
Triangular hebesphenorotunda 13 triangles 3 squares 3 pentagons 1 hexagon

Next up was the triangular hebesphenorotunda, J92. What a beauty! I hadn’t expected to pick favorites, but, as these shapes have so much personality, I guess it was bound to happen.

Feeling bored by having used the watercolor wash papers so much recently, I decided to have a bit of fun with the making a different paper for this shape. Used the flares option in Adobe Illustrator to make a planetary looking paper. Added some hand drawing to one of them. The white one has the pattern inside, which shows up when it’s illuminated from within.

Putting LED tea lights into these shapes was fun. I need to find a good source of bright LED lights good for this purpose. Please pass on any suggestions to me please.

Sphenocorona 12 equilateral triangles and 2 squares

Third shape that made it on to my desk was J86, the sphenocorona, which quickly emerged as one of my favorites. Some of these structures seem to be more willing to engage than others.  The sphenocorona checked just about all my boxes. Its net is pretty, and lends itself well to being collapsed (see pink, left above) then rotated the centrally collapsed net into a snowflake-like structure.

Sphenocorona on a book given to me and my sisters by our Uncle Joe

I chose to put some designs from Islamic Geometry on the sphenocorona. This made me start thinking about why it was I tended to use these geometries on solids, rather than as wall hangings. I wondered if it had to do with my first exposure to Islamic Geometry, which was in this book showing the walls of the Alhambra, which had a place of honor in my mother’s house.

12 equilateral triangles and 2 squares,
Sphenocorona,12 equilateral triangles and 2 squares

Here’s two cool attributes of the sphenocorona. From two different angles they look like two different shapes. Also, this one will make a great cat-friendly Christmas ornament once a string is attached.


Encouraged by how well geometry patterns looked on my shapes, this sphenomegacorona was the thusly adorned.

The sphenomegacoron, J88, with its 16 triangles and two square, might be my darling of the shapes. Feels good, builds well, has a great look to it, plays well with others. By now, though, it was getting later in the month, which seemed unfortunate because I knew that the last week of October wasn’t a going to be much of a hashtag week.

What’s even worse than that though was the next shape, which might have been my last. This shape started out difficult, I lavished attention on it, but nothing worked.

augmented sphenocorona
Augmented sphenocorona 16 triangles, 1 square

The augmented sphenocorona, J87, nearly did me in. I gave it all sorts of attention, put a really cool pattern on it, made lots of them…

augmented sphenocoronas
augmented sphenocoronas16 triangles, 1 square

…really tried to play with them, but it was like they wanted nothing to do with me. Kept working with them until it was time to leave October behind. Thought this would be the end of it.

When I returned home, November came. I didn’t want to end the project on such a bad note. Thought maybe I’d do the last two of the “others” Johnson solids quickly.

Just picked a stock pattern in my Illustrator program and slapped it on to the bilunabirotunda, J91.

Now, before going any further, stop and look at that name. Bi luna bi rotunda. What a pretty name. I feel guilty for having just slapped this one together.

Bilunabirotunda 8 equilateral triangles, 4 regular pentagons, 2 squares

What a charmer! Builds really well, in many different ways. Makes a great snowman, plays well with others, feels stable in numerous placements. Wish I had taken the time to pick out a more thoughtful covering. So silly that I felt rushed just because it wasn’t October any more.

One more shape to go. I took my sweet time with it.

How sweet it was.

Here’s the last others one.

The gorgeous J90 disphenocingulum.

Disphenocingulum 20 equilateral triangles and 4 squares

After experimenting with so many different patterns, it finally occurred to me that curves-of-pursuit patterns would work brilliantly on these solids.

Disphenocingulums with net

Here’s a shape that has such a lovely, rich presence, and which are so darn enchanting that I refused to get irritated at them for not being good for building, and for being soooo other. More than the other others shapes they looked like rocks. So I took them outside.

I let them commune with nature…

…let them fill in the cracks of an old barn.

Reminded myself, if you love something, sometimes you have to let it go. So I let one go. Here it is floating away…

That’s the end of this post. Now for a family photo.

Angela at the Ocean
Angela at the Ocean

Oops. Wrong family photo. That’s my daughter and where we were at the end of October. Exit 0 in New Jersey.

Family photo of Johnson Solids, Others.

Front row, left to right, J87 augmented sphenoncorona, J86 sphenocorona, and  J88 sphenomegacorona,

Back row, left to right, J89 hebesphenomegacorona, J90 disphenocingulum , J91 bilunabirotunda, J92 triangular hebesphenorotunda.

That’s all folks. G’night!

geometry and paper · Math with Art Supplies · Working with Paper

Making Blocks from Paper

There’s a rabbit hole of of making 3D shapes that I’ve stumbled into…

This week I have the good fortune of being able to talk to people, via zoom, at the German Mathematical Society’s 2021 Minisymposium “Mathematics and Arts, about the gyrobifastigium polyhedron. Since there are specific ways solid shapes are talked about it was suggested that I define some of the classifications. No fun in that, unless I can manage to physically toss around some of the supporting actors. There is a certain exclusive club of shapes that my shape belongs to, so I got to work on creating a few samples

My little quest started out as a disaster. The polyhedron friends of my shape looked incredibly shabby.

This story has a happy ending. Things are looking good. Here’s what changed.

This one photo shows more than it lets on, about paper, tabs, and designs.

First the paper. Tried out all sorts. Hands down winner was Hammermill Premium Cardstock, 110 lb, 199gsm.

Notice the tabs on my pattern for the shape. (I’ve learned to call these shapes the net of the shape.) Unlike anything you will find on the sadly misled internet images for nets, putting a generous tab on each raw edge elevates the final product into the upper bracket of awesomeness.

Finally, I needed a dash a beauty that came easy. Messed around in Photoshop, and made some dreamy washes.

Every single fold line has a score line pressed into it – I used a ball point to press in the score line, which is nice because I could see where I had already scored. Until the ink ran out. But that was okay. I managed.

Now this is important. I prepared for the gluing by making those tab folds really sharp, pressing down with a bone folder or the back of a scissors, which ever I found first. This helped make the edges of my shape look and feel really really good.

I used straight PVA for an adhesive on most the tabs, applied thinly with a flat brush. I tried out Elmer’s School Glue. Didn’t like it at all because it took too long to dry. My patience has limits.

The results, finally, were glorious.

I actually couldn’t sleep one night this week because I wanted to play with my new shapes. Finally came back downstairs and messed around with them for a while. Then I could sleep.

Now I will tell you that you didn’t have to read anything I wrote above because it’s all here in a video.

Now if you must start making your own nets, there’s a very cool video that you can watch next, made by my math friend Mark Kaercher, who just happened to post this about this earlier today. Gotta love these synchronicities.

Here’s my little exclusive club of shapes, the cube, the hexagonal and triangular prisms, the truncated octahedron, and, my hero, the gyrobifastigium. What these shapes have in common is that they are the only five regular polyhedra that can tile space. An exclusive club indeed.

One thing kept bothering me about these shapes. I could make them all play together nicely, with the exception of the truncated octahedron, because it was just so darn big. You see, I wanted all the edges of all the shapes to be the same sides, and since the truncated octahedron has so many faces, it just turns out to be really big. Does not play well with others…

Then it occurred to me that maybe…maybe….

…the four other shapes might like to be stored inside the big one.

Oh, yeah!

There they are, one set, all tucked away.

Flexagons · geometry and paper

Flexagons and Me at MoMath

Some time ago I created this new kind of flexagon. I’ve kept the secret to myself long enough that I’m ready to share it. The Museum of Mathematics is hosting my reveal through their events program, via zoom, this coming Thursday, August 12 at 6::30. I want everyone to join me there.

There are many different kinds of flexagons. All of them can be folded into different configurations to reveal hidden sides and to show different patterning. I am enthralled by this genre of paper folding.

Some flexagons are quite tricky to fold. I think of this one that I’ve made as being elegant because it’s super easy to fold. It’s one of those things, that after you know it, you wonder why hadn’t you figured this out yourself.

I want everyone who likes my work to sign up for this event! It’s only $10 to join, and you can have your friends and family sitting with you with the same link. What’s even better is that proceeds will be benefiting the museum itself.

I’ve wanted to do this at MoMath because, throughout the lock-down, I’ve been so impressed by how MoMath kept up their programing via zoom, with programs for everyone from young children to senior citizens. The wide range of people they supported with their programs was truly impressive, and quite affordable including many free events, and lots of $5 sale events.

Oh, one more thing.

I’m making a graphic available for people to decorate. I designed this graphic specifically for my flexagon. I’m not sure if the museum will be distributing it (I think they will), but if not, I will make it available directly for people who sign up.

Gather together a couple of sheets of regular copy paper, glue stick, scissors, scrap paper, and four paper clips.

Be there or be square.

It would mean the world to me.

geometry and paper · Math with Art Supplies

Revisiting the Gryobifastigium

A couple of years ago I wrote about a shape with the wild name gyrobigastigium. I made and sent off a a collection of them to a group show. After they were returned they were boxed up and and tucked away. Recently I was feeling a bit of sadness that all the thoughts and ponderings and joy that this shape brought to me had also gone into deep storage.

I found the box, opened it, and began to move the solids around.

Many gyrobifastigium

These solids were as every bit as engaging as I had remembered them to be.

A few days later I saw, on Twitter,a thread written by Dr, Martin Skrodzk announcing that the German Mathematical Society had put out a call for papers for an on-line mini symposium called “Mathematics and Arts.”

Having just been forlorn, thinking about my shapes, I decided to submit. I had to write an abstract. I’m not really sure what mathematicians want to see in an abstract, so I asked some questions and wrote something.

My submission was accepted!

The symposium happens between Sept 27 and October 1. I think I saw, somewhere on their website that registration is free for teachers. Not 100% sure of that, but am checking. 

If you are interested, here’s the PDF of what I wrote. It’s a short, fun read. I hope you take a look.