Convex Polyhedrons Built of Rhombuses.

We’ll present 4 types of polyhedrons satisfying the above condition.

Type 1. Polyhedron of 20 faces.

We list this type first, because a picture of it is above. For a better understanding we present two more pictures – front and back:

It is easy to see that in each picture there are 5 inner rhombuses and 5 outer ones. Inner rhombuses have angles 30 and 120 degrees (they are the sum of 2 equilateral triangles).

Outer rhombuses have angles 108 and 72 degrees. Angles of inner rhombuses were set by ‘Literka’.

There are only 2 vertices, which are the common points of 5 edges.

This type of a polyhedron can be deformed by changing angles of rhombuses. We can derive a polyhedron built of 20 congruent rhombuses. See a page of ‘Literka’ Few Remarks on Polyhedrons Built of Rhombuses for pictures and description.

Type 2. Polyhedron of 30 faces.

This is a polyhedron built of 30 equal rhombuses. See a special page of ‘Literka’ Polyhedron built of 30 equal rhombuses with few remarks about this polyhedron. Notice that there a similarity between this and previous example. The difference is that we started building previous example with rhombuses of 60 degrees and we had to add some other not-equal rhombuses.

Type 3. Polyhedron of 12 faces.

This is a more regular polyhedron. All vertices are common points of exactly 3 edges and all faces are exactly the same. As it is pointed out in the handbook of H. Steinhaus “Mathematical Kaleidoscope”, we can find 8 vertices of this polyhedron, which are the vertices of a regular hexahedron (See Plato's polyhedrons). ). Changing angles of rhombuses of this polyhedron we can receive a polyhedron built of rhombuses with 4 rhombuses being squares. See a page of ‘Literka’ Few Remarks on Polyhedrons Built of Rhombuses for pictures and description.

Type 4. Polyhedron of 6 faces.

This polyhedron is obtained from the regular hexahedron (See Plato's polyhedrons) by turning its edges.

We have presented 4 examples of types of polyhedrons built of rhombuses. The problem to find more is left to the reader.

See similar pictures and comments on the page “Few Remarks on Polyhedrons Built of Rhombuses”

See related topics "Polyhedrons built of equilateral triangles”.

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