3.    Visual and verbal description of the eight Deltahedra  

As we have seen, altogether, there are exactly eight convex Deltahedra. This set includes: 

  • The Platonic solids: The Tetrahedron D4,  the Octahedron D8, and the Icosahedron D20;

  • Two Dipyramids - D6, and D10.   (D8 which is also a dipyramid is not included here because it is mentioned above).

D6 - Triangular dipyramid  - two attached Tetrahedra

(D8 - Square dipyramid  - two attached square pyramids, actually the Octahedron)

D10 - Pentagonal dipyramid  - two attached pentagonal pyramids

And  - three rather more complicated Deltahedra:

3.1    D14 and D16 – Can you tell the difference?

It is quite difficult to distinguish D14 and D16 from each other by their visual images.

Holding their concrete real models and looking at them from various angles does not help much either.

On the other hand verbal descriptions make it quite clear:

  • D14 is a triangular prism with a a square pyramid attached to each of its side faces;

  • D16 is a square antiprism [1] with a square pyramid attached to each of its bases.

3.2 “Seeing” and believing

We observed high school and college students attempting to describe verbally these two solids, after they saw their actual models and various drawings.

The evidence collected suggest quite clearly, that only after one is able to give a coherent verbal description, one reaches an understanding of the structure of each solid, and is able to “see” the differences between the two solids. - - As if these solids leaped into palpable reality, in Davis’s (ibid) language.

 

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[1]        An antiprism has two congruent regular n-side polygons, as bases, in parallel planes. One base is twisted so that each of its vertices is midway between two vertices of the other, to each of which it is joined. The side faces are triangles. While twisting one of the prism's bases, the square side faces of the prism are folded into triangles, and the result is an antiprism