I would like to thank Adrian Rossiter who wrote the language scripts in his ‘Antiprism’ program (http://www.antiprism.com/index.html) for all of the following polyhedron models that I constructed. I recorded a short video animation of each (listed at the end of this post) using ‘Debut Video Capture’ by NCH Software.
I made the first of these models a while ago using a ‘Jovo Construction Set’… http://www.jovo.com/en_jovo.html. It has eleven faces, comprising five regular hexagons and six trapezia and displays mirror symmetry. The shorter side of the trapezium is half the length of the longer sides. The model is not dissimilar to the regular pentagon and trapezium model linked below… https://hedraweb.wordpress.com/2016/05/02/in-the-beginning/.
More recently, I put together six other cardboard models that utilise a different type of trapezium using the same slot-together method as ‘Itsphun’ shapes (https://itsphun.com/products) but only two were verified by Adrian as being geometrically constructible polyhedra.
The first of these looks like a spinning top. It has twenty faces consisting of twelve golden ratio trapezia and eight equilateral triangles. My model (not shown) had a tendency to only rest on the six trapezium faces that surround the centre triangles. It looks like the two views, below, generated in ‘Antiprism’.
The second of the two models reminds me of a tombola drum. It has twelve faces made up of six trapezia and six rhombi. The model can stand upright on any of its faces.
Here is a clearer presentation of it using Adrian’s ‘Antiprism’ software.
Interestingly, this model has 3D space filling tessellation properties.
Near misses – Adrian confirmed that model ‘B’ was not possible. After further scrutiny of the other three, I noticed obvious flexing of the card, so would not have worked either. A little disappointing really, as they all display interesting symmetries and have both left and right handed versions. A) 12 trapezia/2 triangles, B) 12 trapezia/4 triangles, C) 12 trapezia/6 triangles and D) 12 trapezia/8 triangles.
Animated video clip links…
Eleven faced polyhedron – https://www.mediafire.com/file/9rha41jgcq85shk/hex_trap.mp4/file
Twenty faced polyhedron – https://www.mediafire.com/file/1fzg59pghes5vos/trap_tri_spinning_top_1a.mp4/file
Twelve faced polyhedron – https://www.mediafire.com/file/ljwv7p5wbjqusqq/trap_rhomb_tombola_model1a.mp4/file
Space filling example – https://www.mediafire.com/file/viooi0i4xtqvk4t/trap_rhomb_tombola_space_filler1a.mp4/file