Mathematical Shapes

Welcome to my mathematical shapes blog. Most of what I post will be made by me. These things are time consuming to make so posts are not very frequent. I think my default pic is quite appropriate so I won’t change it. In any case I am far too ugly to post a selfie.

[polyhedronguy]

I had been thinking about this one for quite a while before going ahead and constructing it.

If you take the five platonic solids and arrange them by increasing numbers of vertices, there is an interesting way you can nest them inside each other. You start with a tetrahedron, then an octahedron, then a cube, then an icosahedron, and finally a dodecahedron. With this ordering, you can have each corner of one shape in the precise centre of a face on the next one out.

This model is based on that idea. I made the faces with large parts cut away, but kept the edges intact so that the individual polyhedra are still easy to identify. I also had to come up with a way to attach the vertex of one polyhedron to the centre of the face in the next layer. I went with a system of tabs and patches that is secure and looks tidy. Getting the relative sizes of the shapes right was a fun maths challenge. I used a spreadsheet to calculate the required edge lengths when I had chosen the size of the outer dodecahedron.

I am pleased with the result - you have to peer in at certain angles to be able to see the red tetrahedron in the very middle.

[polyhedronguy]

It turns out there are ten topologically distinct spherical weaves with 24 strands and cubic symmetry. This means that each strand maps to each other strand by a symmetry of the model. There are also ten other arrangements of 24 great circles that do not admit a proper weave. I could not find any literature on spherical weaves so I had to do my own analysis (and I could be wrong!). Please let me know if you know of any existing research in this area. Next, I will look into the weaves with dodecahedral symmetry.

This is the first model I have made with 24 strands. All the symmetry points of the cube can be seen in the photos - the square faces correspond to the six large almost-square octagonal holes, the corners correspond to the eight almost-triangular hexagonal holes, and the edges correspond to twelve squished octagonal holes.

This was not an easy project. I first tried with strips of card, but these were too flimsy to make a self-supporting model. I then tried using a double thickness of card, but that was too difficult to work with. Finally I used chair-cane. This was exactly the right stiffness but it was uneven and produced a model that is a bit lopsided. Oh well, it will have to do for now - I still think it looks cool; it is springy and nice to handle too.

[polyhedronguy]

This the third of three possible 12 strand spherical weaves with cubic symmetry. There are other symmetrical arrangements of great circles but only three can admit a strict weave. The rest have three or more strands crossing at certain points.

This one has eight non regular hexagon holes positioned at the corners of the cube, and six square holes positioned over the cube faces but twisted by 45 degrees.

They are quite difficult to make but perseverance will get you there. The trick is to shorten the strands until they support each other in the desired symmetrical arrangement. Too loose and the model falls out of symmetry. Too tight and the model pulls into a non spherical shape.

Next I will try some 24 strand weaves.

[polyhedronguy]

Here is another symmetrical arrangement of great circles. I have decided to call them "Spherical Weaves". Like the last one, this has 12 strands in cubic symmetry. It has six non-regular octagonal holes.

If anyone knows of a systematic study that has been conducted on these things, I would love to hear about it.

[polyhedronguy]

This is another great circle model with cubic symmetry. It is made from twelve separate strips braided together. It has six octagonal holes aligned with the square faces of a cube, and eight tiny equilateral triangle holes aligned with the cube’s corners.

This was a difficult model to make though not as time consuming as my polyhedra. One thing that made it hard was not being able to find any examples of it online. I eventually did a mockup in GeoGebra.

[polyhedronguy]

I have taken an interest in symmetrical arrangements of great circles, and have started making models of them by weaving strips of thin card. I have devised the following rules for the models:

1) The model must only contain great circles.

2) Any circle can be mapped to any other circle by a symmetry of the model.

3) No three circles may cross each other at the same point.

These rules result in different sets of great circles than those catalogued by Buckminster Fuller (his "25 great circles of the spherical octahedron", and "31 great circles of the spherical icosahedron"). The number of circles in a model ranges up to a maximum of 60 (discounting the infinite family with prismatic symmetry). We shall see whether I succeed in making any of that size!

Any model that satisfies the rules can be constructed as a strict weave (or braid), where the strands alternately go over and under each other. This is related to Alexander's theorem and it is a fun exercise to prove.

The zero-, one- and two-circle arrangements are not very interesting. Note that an arbitrary pair of distinct great circles automatically satisfies all three rules and has D2 symmetry.

The first model worth making has three circles in cubic symmetry. The circles follow the edges of a spherical octahedron. It is interesting that no two circles in the weave are linked even though the complete model is linked. It is topologically equivalent to the Borromean Rings.

Next up is the four circle model, also with cubic symmetry (the blue strips in the pictures). In this case the circles follow the edges of a spherical cuboctahedron. Finally, I made a six strip model with pyritohedral symmetry (the red strips). I had no idea this arrangement existed until I started my research. I combined the red and blue strips into a single arrangement because it made for a more attractive model. The combined model is very close to (but not quite the same as) a ten-strip model with dodecahedral symmetry.

[polyhedronguy]

This shape is a snub icosidodecadodecahedron. It has twelve pentagram faces, twelve pentagons, and eighty triangles. The triangles are all the same size of course, but they can be divided into two groups - twenty icosahedron faces all coloured orange and sixty snub faces in various colours. The snub faces form very sharp reflex angles with the pentagram faces. It is not very clear from looking at the finished model, but around each star face are five deep but very narrow cavities formed entirely from facets of the snub faces.

Snub polyhedra can be constructed by starting with a truncated shape (in this case an icositruncated dodecadodecahedron), drawing edges alternating around its faces, and then adjusting the edge lengths to make everything regular. It is very difficult to visualise the relationship even when the two shapes are shown side by side.

I chose to colour the faces using six colours for the faces in dodecahedron planes (stars and pentagons), and a seventh for the icosahedron triangles. I was then able to use a symmetrical arrangement for the snub faces so that adjacent faces were different colours.

This was not the hardest shape I have made but it was still very difficult to assemble and was more challenging than I expected. As with many polyhedra with this complexity, I enjoy looking at it from different angles to see the various symmetries.

[polyhedronguy]

This is a Snub Dodecadodecahedron, made from 12 pentagrams, 12 pentagons and 60 triangles. I made it in six colours, with the pentagon/pentagram faces in parallel planes having the same colour. I managed to colour the triangle faces so that adjacent facets never have the same colour. As usual, I made a detailed colour-plan before I started assembly.

The most remarkable feature of the shape is that it has facets that are very thin slivers. Many representations you might see online are not accurate enough for you to make them out, so I included a close-up to illustrate how intricate they are.

Not surprisingly, this was extremely difficult and time consuming to make. The result is not the standard I was aiming for, but I still like the finished piece. The overall effect is slightly unsettling, as if the polyhedron is fractured, and not quite the right size for itself.

[polyhedronguy]

This chaotic looking object is something I designed, based on the small snub icosicosidodecahedron that I made recently. Like it’s counterpart, it has twelve pentagram-type pieces and one hundred triangle-type pieces. I used cutaway faces with petal-shaped hooks at the corners. They were designed so as to veer around each other and form a rosette at each vertex, so no glueing was necessary.

In the “normal” version, there are twenty skew hexagram faces formed from pairs of triangles. In this version I was able to use a separate piece for each triangle in the hexagram, so they could have separate colours. It turns out that each of the twenty possible pairs of different colours is represented in the finished model. I decided to fold a crown-shape on the lower piece through a hole in the upper piece, which helped to secure the pieces together as well as look cool.

Assembly took much less time than for the normal version, but the design phase was more difficult. I had to do some maths to determine the angles of the petal-hooks, so that they ended up evenly spaced around the vertices in the finished model. The sixty snub pieces are not symmetric, but are difficult to orient by eye, so I put a teardrop-shaped hole in them so I wouldn’t get confused.

As usual with this style of model, the interior is very attractive too. Fortunately it is still visible through the numerous holes.

[polyhedronguy]

This is a small snub icosicosidodecahedron, with twelve pentagram faces and 100 triangles. There are twenty pairs of triangles that overlay each other at skew angles to form irregular hexagrams in the planes of an icosahedron. The remaining sixty triangles are the snub faces, and they don’t share any symmetries with the polyhedron they are part of. Although this shape is non-convex, it is very close to spherical, and the angles between adjacent faces are extremely shallow.

Unusually for a snub polyhedron, this shape has mirror symmetries.

I chose, an icosahedral colouring pattern using five colours plus a sixth colour for the star faces. It turns out that I could select colours for the snub faces so that they followed the map colouring principle (adjacent regions have different colours). Because the different kinds of facets are so similar, I gave them distinctive tabs to avoid mistakes during assembly. Preparing a detailed colouring plan in advance was also essential, but after that construction was straightforward. I was very pleased with the result.

[polyhedronguy]

Compound of five octahedra.

The third stellation of the icosahedron is essentially the same as a compound of five octahedra. However, most models don’t address the issue that there are 5x8=40 octahedron faces in the compound and only 20 faces in the stellation. This is explained by noting that the triangular octahedron faces come together in pairs, but of course that means that the colour of each stellation face is ambiguous. A solid model can avoid the issue because the visible outer corners are distinct and can have separate colours. With this model, I have used cutaway faces, so the pairs of faces must overlap each other.

By using two different cutting patterns, alternating over each octahedron, the model can be assembled without the pieces interfering with each other. I used differently sized central holes so that the pairs of colours can be seen easily. It turns out that every ordered-pair of colours is represented in the completed model.

As usual, the interior view is very attractive, but it is sadly not visible in the finished piece. I like the overall effect - it reminds me of Celtic knot-work.

[polyhedronguy]

This shape is a Great Rhombidodecahedron. It has twelve decagram faces and thirty squares with intersections that are difficult to get your head around. The facets vary a lot in size ranging from the large chevrons to the tiny triangles in pairs at the base of the diamond cups.

There are many deep concavities, including twelve cups with parallel edges going up the sides. It turns out that each decagram face is perpendicular to ten of the squares (though not the squares' edges). This suggested a colouring scheme - I grouped the squares and decagrams into six different coloured sets of two decagrams and five squares. The squares in each set are perpendicular to the decagrams. By doing this I ensured that very few facets are adjacent to other facets of the same colour. To get everything right I had to prepare a detailed colouring plan before I started gluing.

As usual for a polyhedron this complex, it was fiddly, time consuming, but also very rewarding to make. With the completion of this shape, I have now made each of the 75 uniform polyhedra at least once, apart from the non-convex snub polyhedra. That leaves me ten more, including some real monsters! I guess it will take me a long time before I finish the set.

[polyhedronguy]

This shape is a great dodecahemidodecahedron. It has twelve pentagrams and six decagrams in six sets of parallel planes; each plane has its own colour. It is closely related to the great icosahemidodecahedron that I posted recently, having the same vertices and edges, and decagrams that go through the centre of the shape.

I constructed it in much the same way as the other shape, making the concave sections and then assembling them. It was also quite fiddly because they were tightly packed on the inside.

I like how the colours look completely different at different orientations.

[polyhedronguy]

This is a Great Icosahemidodecahedron. It has twenty triangles in five colours and six blue decagrams. The planes of the decagrams all pass through the centre of the solid.

I made it by first gluing the 12 rosettes of diamonds and the 20 blue cups with coloured facets at their base. Then I assembled the components together. On the inside they fit very tightly with barely enough room for the tabs. This made an otherwise straightforward construction a bit fiddly.

I think the result looks a bit like a ball of flowers.

[polyhedronguy]

This is a Quasirhombicosidodecahedron, with twelve green pentagrams, twenty cream triangles and thirty purple squares.

It is quite intricate, with nine edges crisscrossing around each of the triangular cups. Wenninger described it as “not too difficult to make” but “requiring much perseverance“. I guess I would agree, but the sheer amount of work means it is not for the faint hearted!

The finished model looks very impressive, with visual impact from the star faces or the triangular faces depending on the orientation. I also liked how the purple sections looked like bats before I glued them in.

[polyhedronguy]

This is a great dodecahemicosahedron. It has twelve pentagons in six colours, and ten hexagons in green. The hexagons all go through the centre of the shape.

I made it by assembling twenty corner sections and twelve star-shaped sections, then glueing these together. It was quite straightforward to make, and very satisfying. I like the deep pentagonal cups with the coloured facets at the base.

[polyhedronguy]

These models have rigid struts that are connected to each other using only tension. None of the struts touches any of the others as the elastic pulls them apart. They are sometimes called tensegrity structures but they don’t meet the strict definition. This is because the struts are not only connected at their ends and so they are not in pure compression.

The five models are my own design, based on the five Platonic solids. They are made from coffee stirrers and small elastic bands used in hairdressing. They are considerably quicker and easier to make than my card models.

Each model is highly symmetrical though the sticks are at skew angles to each other. The elastic in them makes them nice to handle, and they bounce if you drop them on the floor (something I would never dare to do with my card models).