This page and the models are generated by a Python script [MTUN3], which requires the package [MTUN1]. The models are displayed with help of a simple OFF file viewer written in JavaScript [MTUN2].
Note: for optimal experience Javascript should be enabled.
Introduction
During fall 2025 the Mason Green's and his fascinated me and I wondered whether I could use some general rule, other than Mason's rules. to generate more of these. The polyhedra describe here I called roof top polyhedra, and they are all nn3-acrohedra, where n is a rational number.
The main inspiration for the research came from [MCNE1], There it is stated that a 993-acrohedron hasn't been found yet. I didn't have any illusions that I would find one, but it caught my interest and I became curious about what the problem could be. I wanted to build up some understanding. I did have contact with Jim McNeill and he send me some explanation, but I started with ignoring those comments, since I was afraid that it would lead me into a direction that had already been investigated.
My first attempts went in to another direction than what is discussed here. After some more contact with Jim I, used the following way of interpreting the small and great supersemicupola. They both consist of sides that can be interpreted as roof tops, where two congruent faces share an edge and to both vertices equilateral triangles are added. Then these roof tops are connected to a base with n-fold symmetry and these roof tops are orbited according to this rotational symmetry.
The fascinating thing is that for the Supersemicupolae everything magically fits. I wondered whether there are more cases. I wondered specifically how this looked for the enneagon, the 993 case.
Obviously it doesn't work for the enneagon otherwise Mason Green would already have found it. Not surprisingly for the enneagon it just doesn't work. This is how a can look like. Note that I only show one side of the roof. In the case of n=7 they connect edge-to-edge to the next ones. If you do add the other side here, then you will end up with even more loose edges that don't fit. There are more cases where these roof tops just don't fit. This makes it all the more special that it does work for n=7.
When it does work out some surprising polyhedra can be found. My friend Don Romano quickly build one that uses pentragrams as the roof top sides as shown in the image below. It is a model of this one: .
The interesting effect here is that one triangle side of the roof top ends being parallel to the base polygon and all three coincide. I.e. this is an octahedron consisting of three pentagrams and 5 equilateral triangles.
Initial Setup
These roof top polyhedra have an axial symmetry and the sides consist of regular {m/q} polygons and equilaterial triangles. Two of these {m/q} polygons and two triangles form a roof top as follows.
Take a pair of {m/q} polygons let them meet in one edge. Let's define that vertex number 1 of both sides coincide with each other and vertex number n of both sides as well. The dihedral angle between these two polygons is such that the distance between vertex number 2 of both polygons is one edge length and therefore one can fit exactly an equilateral triangle at vertex no. 1 and vertex number 2, and these four polygons form a roof top. For example consider the and the . Note that for m=3 the roof top will be a tetrahedron; therefore it makes sense to require m > 3.
The side of a roof top from the {m/q} polygon can be attached to a regular base polygon {n/p}. Let's assume that it is the triangle edge that connects vertex number 2 from both of the {m/q} polygons. Also assume that vertex number 2 of side number 1 is attached to vertex number 1 of the base polygon and vertex number 2 of the side number 2 is connected to vertex number 2 of the base polygon.
The dihedral angle between the base polygon and the triangle is such that the distance between vertex 3 of side 2 and vertex 3 of the base polygon is exactly one edge length.
This means that the following three vertices are exactly one edge distance from each other:
- Vertex number 2 from the base polygon, which is the same as vertex number 2 from side number 2.
- Vertex number 3 from the base polygon.
- Vertex number 3 from side number 2.
This means these form an equilateral triangle, but it isn't added to the object yet. This is discussed further somewhere else. This shows an example of the right angle. For clarity reasons this example also shows the extra triangle that is not supposed to be added in turquoise.
In general there are two different fold angles that lead to extra triangles that are equilateral. They can either be on the same side of the base as the roof top triangle that is attached to the base, or at the opposite side. These will be indicated by 's' for same side and 'o' for opposite side. In the example above they are on the same side.
The next step is to copy the roof top to all the other edges by rotating them around the symmetry axis of the base polygon by steps of 1/nth turns.
The Finishing Touch
The method described above generally doesn't lead to a polyhedron, but for Mason Green's supersemicupola something remarkable happens. When rotating one side of the roof top is mapped on the other side of the roof top. Clearly this wouldn't happen for the . One can see that both triangles have different dihedral angles with the base. If however you start with an regular heptagon roof top and attach it to {7/2} the small supersemicupola is obtained. Note that only need one side of the roof top when you rotate the roof tops around the symmetry axis. The same happens for a {7/3} base with {7/2} roof tops, which generate the great supersemicupola.
For most other case we are left with an object that has loose edges and it still needs to be closed. Here only regular polygons are considered. The following options exists to make a polyhedron out of the shape that is obtained when the roof tops are copied by rotating around the n-fold axis of the base.
In some case to turn the shape into a polyhedron one {m/q} polygon of the roof top needs to be removed. As stated this is what happens for the . Sometimes even the triangles are mapped onto each other, e.g. for the , where all triangles that don't share an edge with the base polygon coincide.
If there are loose edges it might help to add the extra triangles. When this is done it is often best to remove the base polygon to prevent having three faces sharing one edge. This can be done for rectangle roofs on top of an octagram {8/2} both with , and . On this page the extra triangles are shown in turquoise.
Quite often that isn't enough since the roof triangles that don't share an edge with the original base polygon still have loose edges. In the example above they weren't loose because these edges meet each other. When these edges are loose they can sometimes be connected by adding an alternative base. This is only possible when the value 'm' in the {m/q} polygon is even. Examples of this are the rectangle roofs on top of an octagon both with , and . Note that in both examples the alternative base isn't an octagon but a {8/2} octagram.
Above it is mentioned that it is best to remove the original base polygon when the extra triangles are added, otherwise three faces share an edge. It is however possible to just add one extra base polygon that coincides with the original. Now you would have 4 faces sharing one edge, but you could split these and interpret these as separate edges that happen to end up at the same position. This would be an acceptable way according to [BGRU1].
E.g. if you add the extra triangles and remove the original base and add an alternative base for square roof tops on an octagram {8/3} with the extra triangles on the same side, then it looks like . While this looks like a nice polyhedron it would be impractical to build a model because of the acute angles at which faces meet and pairs of squares that are almost coplanar. Instead you could add the extra triangles and add the alternative base as before, but keep the original base and add another copy. In that case the model would look like , Note that in this case there is more to this model than meets the eye. The abstract polyhedron has eight triangles inside and two {8/3} octagrams that coincide, Since this is invisible the OFF file doesn't contain a definition for them.
Sometimes other options might exist. If for example squares roofs on a triangle, then on the inside the vertices are those of an octahedron. The extra triangles and the alternative base are faces of that octahedron. It is possible to use squares instead as well. This is how the polyhedron looks if you finish the shape in , as described above and this is how it looks , by adding purple-blue squares instead of triangles. However what is really done here is that the bottom of the roof is added and the polyhedron becomes a compound, which isn't really that interesting here and will not be considered further.
To summarise the ways to finish the shape can be recognised:
- Only use one side of the roof.
- Only use one side of the roof and remove triangles.
- Add extra triangles and remove the base polygon.
- Add extra triangles and add extra base polygon.
- Add extra triangles, remove the base polygon and add alternative base polygon.
- Add extra triangles, add extra base polygon and add alternative base polygon.
- Any alternative other way that isn't described here.
The table below summarises some polyhedra that were found with square roof tops up to dodecagon / dodecagram base. The buttons in the left column will show a 3D model on this page, the buttons in the right-most column will copy a link to you clip board. This link will open the model in the OFF file viewer att this site.
The addition 'same side' means that the extra triangles are on the same side of the original base as the roof triangles that are attached to the base polygon. The post-fix '- opposite' means they are on the opposite side.
Similarly the term 'base triangles' refers to the roof top triangles that are attached to the base, while 'opposite triangles' refers to the other roof top triangles. Remove repeated triangles means that triangles that some triangles end up at the same location and that one should delete all but one.
Note that with square roofs it is possible to add an extra square to close the roof. In that case a triangular prisms is obtained and the polyhedron will be a compound of prisms instead.
As an example consider this . This is really a compound of 4 triangular prisms. One where edges conincide.
The compound does look like an interesting model, but compounds aren't considered here.
| Base Polygon | Steps to Turn into Polyhedron | Link to Model |
|
||
|
||
|
||
|
||
|
||
|
||
|
||
Note: Complicated inner structure. |
||
|
||
|
||
|
||
Note: Complicated inner structure. |
||
|
||
|
||
|
||
|
||
|
||
|
||
|
||
Note: Complicated inner structure. |
||
|
||
Note: Cave with tight entrance. |
||
|
||
|
||
|
||
Note: Complicated inner structure. |
The table below summarises some polyhedra that were found with pentagon roof tops up to dodecagon / dodecagram base. The buttons in the left column will show a 3D model on this page, the buttons in the right-most column will copy a link to you clip board. This link will open the model in the OFF file viewer att this site.
The addition 'same side' means that the extra triangles are on the same side of the original base as the roof triangles that are attached to the base polygon. The post-fix '- opposite' means they are on the opposite side.
Similarly the term 'base triangles' refers to the roof top triangles that are attached to the base, while 'opposite triangles' refers to the other roof top triangles. Remove repeated triangles means that triangles that some triangles end up at the same location and that one should delete all but one.
| Base Polygon | Steps to Turn into Polyhedron | Link to Model |
Note: This is the tridiminished icosahedron, a Johnson solid indicated by J63. |
||
Note: The shared edges between the triangles should be seen as separate from the shared edges between the pentagons that coincide. It makes more sense to use the alternative in then next row. |
||
Note: Technically not a roof top anymore. Mentioned by [MCNE1] as a 553-acrohedron. |
||
Note: Compound of two J63. |
||
Note: Compound of three J63. |
||
Note: The opposite triangles end up completely inside. The alternative base falls into the same plane as two extra faces. These faces look like pentagrams, but they are actually concave decagons, otherwise there will be loose edges. It means that they aren't really regular polygons anymore. It makes more sense to remove the opposite triangles and the extra base. This option is added separately in the next row, but on the outside everything looks the same. |
||
Note: Technically not a roof top polyhedron anymore. The extra faces aren't two pentagrams they are actually concave decagons.It means that they aren't really regular polygons anymore. |
||
Note: Compound of two {5/1} roofs on {5/1} base. The same remark holds here, it make sense to remove the triangles and the base as shown in the next row |
||
Note: Technically not a roof top polyhedron anymore. Compound of the two 553-acrohedron as mentioned by [MCNE1]. |
||
Note: Compound of three J63. |
The table below summarises some polyhedra that were found with {5/2} pentagram roof tops up to dodecagon / dodecagram base. The buttons in the left column will show a 3D model on this page, the buttons in the right-most column will copy a link to you clip board. This link will open the model in the OFF file viewer att this site.
The addition 'same side' means that the extra triangles are on the same side of the original base as the roof triangles that are attached to the base polygon. The post-fix '- opposite' means they are on the opposite side.
Similarly the term 'base triangles' refers to the roof top triangles that are attached to the base, while 'opposite triangles' refers to the other roof top triangles. Remove repeated triangles means that triangles that some triangles end up at the same location and that one should delete all but one.
| Base Polygon | Steps to Turn into Polyhedron | Link to Model |
Note: The roof triangles not attached to the base become parallel to the base. |
||
Note: Technically not a roof top polyhedron anymore. Mentioned in [RKLI1] |
||
Note: Compound of two roof top polyhedra with pentagram sides on a triangle base. |
||
Note: Compound of three roof top polyhedra with pentagram sides on a triangle base. |
||
Note: Two edges of the light-green triangles are shared with a pentagram. I.e. the pentagrams don't share an edge, neither do the light green triangles. They just end up at the same spot, but they should be seen as separate edges. Furthermore extra decagrams are needed and these fall into same plane as the alternative base consisting of two pentagons, and for that reason that cannot be recognised here. It makes more sense to remove the light-green triangles and the extra two pentagons as shown in the next row. There you can see the decagrams. |
||
Note: Technically not a roof top polyhedron anymore. The two faces that look like pentagrams are actually (self-)intersecting decagrams. They must be to make sure faces meet edge-to-edge. |
||
Note: Technically not a roof top polyhedron anymore. This is a compound of two polyhedra based on pentagram roof top polyhedra on a pentagram base |
||
Note: Compound of four roof top polyhedra with pentagram sides on a triangle base. |
The table below summarises some polyhedra that were found with heptagon roof tops up to dodecagon / dodecagram base. The buttons in the left column will show a 3D model on this page, the buttons in the right-most column will copy a link to you clip board. This link will open the model in the OFF file viewer att this site.
The addition 'same side' means that the extra triangles are on the same side of the original base as the roof triangles that are attached to the base polygon. The post-fix '- opposite' means they are on the opposite side.
Similarly the term 'base triangles' refers to the roof top triangles that are attached to the base, while 'opposite triangles' refers to the other roof top triangles. Remove repeated triangles means that triangles that some triangles end up at the same location and that one should delete all but one.
| Base Polygon | Steps to Turn into Polyhedron | Link to Model |
Note: Mason Green's small supersemicupola. |
||
Note: Compound of two Mason Green's small supersemicupolae. |
All OFF files that are use on this web page can be found here. However this web site uses a simple OFF file viewer [MTUN2] that has problems with showing self-intersecting polygons. Therefore the OFF files are adjusted so that the outlines of the polygons are shown. This means that edges will be split. I understand that if you would like to view these in Stella then it will generate an error that edges have an odd amount of faces attached.
Besides that to show the models in some interesting angle, The script that generates them uses the X-axis as the n-fold axis. After that I rotate them 105° around the X-axis first, then -20° around the Y-axis. Here I follow the convention that the X-axis points towards the left, the Y-axis points upwards, and the Z-axis points towards the viewer. This means that the polyhedra have some odd orientation.
First of all thanks to Jim McNeill for the inspiration and discussion. I am sorry that I don't take so much time to return the favour. Also thank you Don Romano for fruitful discussions and pointing out mistakes. Thank you Ulrich Mikloweit and Piotr Pawlikowski for listening to me and accepting my spamming of emails, when I think I found something, correcting it, and going back and forth.
References
- [BGRU1] : Are Your Polyhedra the Same as My Polyhedra? Branko Grünbaum; Discrete and Computational Geometry, Springer, New York 2003, pp. 461 – 488
- [MCNE1] : Interactive Polyhedra, Jim McNeill; e.g. about nn3-acrohedra
- [MIKL1] : Ulrich Miklweit's Polyhedron Garden, web site; Available here
- [MIRA1] : Miraheze MediaWiki with the Great SupersemicupolaCupolae, miraheze; Available here
- [MTUN1] : The script to generate the models requires the Python package Orbitit, github; Available here
- [MTUN2] : Interactive 3D models through JS package show-off, github; Available here
- [MTUN3] : All models created with Python package orbitit-scripts, github; Available here
- [RKLI1] : Axial-Symmetrical Edge-Facetings of Uniform Polyhedra, Dr. R. Klitzing; Symmetry: Culture and Science, Springer, Vol. 13, No. 3-4, 2002, pp. 241-258
- [WEBB1] : Stella: Polyhedron navigator, Robert Webb; Available here
- [WIKI1] : Wikipedia: Polyhedron