
two pentagonal "drums"
prism truncated through the midpoints of the edges antidiamond truncated through the midpoints of the edges 

two "drums" with rhombic and regular faces (Jim McNeill)
six faces of this "hexagonal drum" are SR2 rhombi ten faces of this "decagonal drum" are golden rhombi 


On the left a cuboctahedron has been cut in two , orthogonally to a 3fold symmetry axe (the equatorial section is a regular hexagon), and then one part has been rotated by 60°; this assembling of two triangular cupolas is the Johnson's polyhedron J27.
On its dual, the rhombic dodecahedron, the same handling leads to a trapezorhombic dodecahedron (six rhombuses have been changed into a ring of isosceles trapeziums). 

one of the archimedean solids: 
These two polyhedra have the same regular faces (8 equilateral triangles and 18 squares) and all vertices identical; they are both assemblages of two square cupolas (Johnson J04) on the bases of an octagonal prism... They are nevertheless different!
We notice quite easily that the one on the left is more aesthetic  in fact "more symmetric"  than the one on the right (for example only the one on the left has a center of symmetry). Here the important fact is that each vertex plays the same role in the polyhedron as a whole. Miller's solid  or Johnson J37 has only one "equatorial" ring of 8 squares (and not three). Remark: the Miller's polyhedron, sometimes called pseudorhombicuboctahedron, is also attributed to Achkinouze and Bert. 
discovered by J. C. P. Miller... 
Their duals have the same 24 identical kite shaped faces but with two different arrangements.
These two polyhedra can by cut into two identical parts along an "octagonal equator"; thus we can switch from one to the other by a 45° rotation of the two "hemispheres". 
Remark: this rotation technique of a part obtained by a cut containing a "ring of edges" (regular polygon) can be applied to other Archimedes' polyhedra; so we get Johnson's polyhedra (polyhedra with regular faces).
This polyhedron is the dual of Martin Trump's S_{9} (whose nine vertices are "regularly" distributed on a sphere). Three of its faces are rhombuses and the six others are non regular pentagons with a mirror symmetry. 

The midpoints of the edges of a tetrahedron define an octahedron; it's the dual of the parallelepiped in which the tetrahedron is "inscribed". Its edges define three parallelograms and its opposite faces are symmetric with respect to its center (common midpoint of its three diagonals).
This figure may be dynamically modified by moving the vertices of the tetrahedron (yellow points) with the mouse pointer.
see also the generalization of Varignon's theorem 


Here we have truncated the two semiregular rhombic polyhedra at their vertices of highest order.
Don't mistake the second, which is the structure of the C80 fullerene, with the truncated icosahedron (structure of the C60). 4truncated rhombic dodecahedron
5truncated rhombic triacontahedron


More examples on an other page.
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