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!

3 thoughts on “Six Weeks, Seven Shapes

    1. When I took geometry in high school we did lots of constructions with ruler and compass. I loved it. We didn’t make beautiful things, more like just divided lines in half, yet, still, thought being Ble to do that was so powerful! I think geometry just suits me. Glad to share the fun

      Liked by 1 person

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