miscellaneous examples of convex polyhedra

Reminder: A polyhedron is convex if all its diagonals are inside or on its surface.
A diagonal of a polyhedron is a segment joining two vertices and which is not an edge.

a few "drums"

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

J27 and its dual

On the left a cuboctahedron has been cut in two , orthogonally to a 3-fold 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 trapezo-rhombic dodecahedron (six rhombuses have been changed into a ring of isosceles trapeziums).

Miller's polyhedron

one of the archimedean solids:
the small rhombicuboctahedron

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 pseudo-rhombicuboctahedron, is also attributed to Achkinouze and Bert.

discovered by J. C. P. Miller...
after an incorrect assemblage!

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).

a polyhedron with nine faces

This polyhedron is the dual of Martin Trump's S9 (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.

Varignon's octahedron

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.
The "f" key is a toggle for the display the faces.

see also the generalization of Varignon's theorem

truncated rhombic polyhedra

Here we have truncated the two semi-regular 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).

4-truncated rhombic dodecahedron
(8+6×4=32 vertices, 12 hexagons and 6 squares)

5-truncated rhombic triacontahedron
(20+12×5=80 vertices, 30 hexagons and 12 pentagons)

Remark: the hexagonal faces of these two polyhedra are obviously not regular; think about three regular hexagons assembled around a common vertex!

More examples on an other page.

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convex polyhedra - non convex polyhedra - interesting polyhedra - related subjects April 1999
updated 08-03-2014