Face Probabilities of Archimedean Solids

In summary, the discussion revolves around the calculation of probabilities for Archimedean Solids landing on a specific face when rolled. The conversation also touches upon whether the edges and vertices need to be taken into consideration or if the face area is enough for a good approximation. It is mentioned that in an ideal situation, the size of the object may determine the probability, but in reality, drag forces and weight would also play a role. The individual also asks for any existing studies or keywords to search for on arXiv.
  • #1
Spinalcold
18
0
I'm looking for a way to calculate the probabilities of Archimedean Solids landing on a specific face if a person would roll one. Of course, not the regular polygons like cubes and dodecahedrons, but something with more than one type of face like the snub cube or truncated icosahedron.

I am wondering of the edges and vertices would need to be calculated as well as face area, or if the area would be enough to get a good approximation of this. If area is the dominant parameter on this it would be interesting that size determines the probability in an ideal situation. Of course, in reality it wouldn't be ideal and drag forces (and weight of the object) would change these probabilities as well.

If this has already been studied (and I'm guessing it likely has been), I'd be grateful for links to papers, or even the appropriate key words to search for on arXiv.

Thanks!
 
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  • #2
https://www.quora.com/How-does-one-calculate-the-probability-that-an-irregular-polyhedron-when-rolled-lands-on-a-particular-face
 
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FAQ: Face Probabilities of Archimedean Solids

What are Archimedean solids?

Archimedean solids are a group of 13 convex polyhedra that have identical regular polygons as their faces. They were named after the ancient Greek mathematician, Archimedes, who first studied them.

What are the properties of Archimedean solids?

The properties of Archimedean solids include having identical regular polygons as their faces, having identical vertices, and having identical angles between faces. They also have a high degree of symmetry and can be divided into identical smaller polyhedra.

How are Archimedean solids different from Platonic solids?

Archimedean solids differ from Platonic solids in that their faces are not all identical regular polygons. In Archimedean solids, the faces can be different regular polygons, but the arrangement of the faces and vertices still results in a highly symmetric shape. In contrast, Platonic solids have identical faces, vertices, and angles.

What are the applications of Archimedean solids?

Archimedean solids have many applications in mathematics, engineering, and art. They are used to model and understand complex structures, such as crystals, viruses, and molecules. They also have aesthetic value and have been incorporated into various designs, such as architecture and jewelry.

How can we calculate the face probabilities of an Archimedean solid?

The face probabilities of an Archimedean solid can be calculated by dividing the number of faces of each type by the total number of faces. For example, if an Archimedean solid has 8 triangular faces and 6 square faces, the face probabilities would be 8/14 or 0.57 for triangles and 6/14 or 0.43 for squares.

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