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Take a naive interpretation of regular polyhedra:

All vertices (including epsilon ball) congruent

All edges congruent

All faces congruent

We can now find interesting families by removing one requirement. For example the uniform polyhedra have all vertices and edges congruent, but not all faces, and their duals have faces and edges congruent, but not vertices.

Are there examples, or interesting families, of polyhedra where every pair of faces is congruent and every pair of vertices, but not every pair of edges?

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By "vertices", I assume you mean "vertex links". –  Igor Rivin Sep 13 '11 at 13:42
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Yes, though personally I find "vertex (up to a small ball)" more intuitive. –  Edmund Harriss Sep 13 '11 at 14:11
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3 Answers 3

up vote 11 down vote accepted

It turns out that these polyhedra that have congruent vertices and faces have a name. They are the Noble Polyhedra. If one insists that they also be convex the Noble polyhedra are the regular polyhedra plus the disphenoids mentioned in Douglas Zare's answer.

When one allows intersecting faces, however, new collections turn up, such as the stephanoids, originally studied by Max Brüker:

Max Brückner Uber die gleichecking-gleichflachigen, diskontinuierlichen und nichtkonvexen Polyheder
Nova Acta Leop. 86(1906), No. 1, pp. 1 – 348 + 29 plates.
Images of the plates with pictures of the models.

These shapes are also discussed and further developed by Branko Grünbaum:

Polyhedra with hollow faces
Proc. NATO-ASI Conf. on polytopes: abstract, convex and computational, Toronto 1983, Ed. Bisztriczky, T. Et Al., Kluwer Academic (1994), p 43-70.

Grünbaum's constructions do use generalisations of the definition of polyhedra. For a thorough discussion of these (including having polygons return to the same vertex, and coplanar faces) see the following paper, which also has a discussion of Noble Polyhedra.

Grünbaum, B. Are your polyhedra the same as my polyhedra?
Discrete and Computational Geometry: The Goodman-Pollack Festschrift B. Aronov, S. Basu, J. Pach, and Sharir, M., eds. Springer, New York 2003, pp. 461 – 488.
http://www.math.washington.edu/~grunbaum/Your%20polyhedra-my%20polyhedra.pdf

Interestingly the classification of all Noble Polyhedra is still an open problem.

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@Edmund: Neat! Here is the Wikipedia entry on: en.wikipedia.org/wiki/Noble_polyhedron . –  Joseph O'Rourke Sep 26 '11 at 15:45
    
It does need a little work though. I hope to find time soon to make some computer models and images. Noble polyhedra have barely made it to the web! –  Edmund Harriss Sep 28 '11 at 14:53
    
But if you accept Grünbaum's definition of polyhedron, the classification of all regular polyhedra is also an open problem! –  JeffE May 19 '12 at 9:17
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Are there any resources in English on Brückner's 1906 polyhedra? –  Greg Zitelli Aug 4 '12 at 17:07
    
I would also like to know about any English translation of Brückner's paper, or another English proof that these are the only convex noble polyhedra. –  kundor Sep 7 '13 at 18:39
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Rhombic disphenoids are examples of polyhedra with identical vertices and faces, but distinguishable edges. These are irregular tetrahedra whose faces are scalene triangles. Their symmetry groups are isomorphic to $C_2 \oplus C_2$, and act transitively on the faces and vertices. You can make a disphenoid from an acute triangle by folding along the line segments connecting the midpoints of the sides.

All tetrahedra whose sides have equal area are disphenoids. Also, ideal hyperbolic tetrahedra have the same symmetries as a disphenoid.

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Had hoped for a comment on the existance or impossibility of such shapes beyond tetrahedra. Otherwise would not have delayed accepting the answer so long. Thanks! –  Edmund Harriss Sep 17 '11 at 16:08
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Allan Edmonds has a few papers studying "equifacetal simplices", i.e. simplices in which any two facets must be congruent. Here are references:

Edmonds, Allan, The center conjecture for equifacetal simplices. Adv. Geom. 9 (2009), no. 4, 563--576.

Edmonds, Allan L., The partition problem for equifacetal simplices. Beitrage Algebra Geom. 50 (2009), no. 1, 195-213.

Edmonds, Allan, The geometry of an equifacetal simplex. Mathematika 52 (2005), no. 1-2, 31-45.

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