# Convex Polyhedra with Regular Faces

@article{Johnson1966ConvexPW, title={Convex Polyhedra with Regular Faces}, author={Norman W. Johnson}, journal={Canadian Journal of Mathematics}, year={1966}, volume={18}, pages={169 - 200} }

An interesting set of geometric figures is composed of the convex polyhedra in Euclidean 3-space whose faces are regular polygons (not necessarily all of the same kind). A polyhedron with regular faces is uniform if it has symmetry operations taking a given vertex into each of the other vertices in turn (5, p. 402). If in addition all the faces are alike, the polyhedron is regular. That there are just five convex regular polyhedra—the so-called Platonic solids—was proved by Euclid in the…

## 158 Citations

### Convex regular-faced polyhedra indecomposable by any plane to regular-faced polyhedra

- Mathematics
- 2009

A convex polyhedron with regular faces or with faces decomposable by two regular polygons is called indecomposable if any section plane dissects this polyhedron into two parts so that at least one of…

### Junction of noncomposite polyhedra

- Mathematics
- 2010

All 3-dimensional convex regular-hedra are found, i.e., the convex polyhedra having positive curvature of each vertex and such that every face is either a regular polygon or is composed of two…

### A Polyhedral Byway

- Art, Mathematics
- 2001

BRIDGES Mathematical Connections in Art, Music, and Science An investigation to find polyhedra having faces that include only regular polygons along with a particular concave equilateral pentagon…

### Wythoffian Skeletal Polyhedra in Ordinary Space, I

- MathematicsDiscret. Comput. Geom.
- 2016

The blueprint for the construction is described and the Wythoffians for distinguished classes of regular polyhedra are treated, which are vertex-transitive and often feature vertex configurations with an attractive mix of different face shapes.

### Regular Polyhedral Clones

- MathematicsThe Mathematical Gazette
- 2013

For any polyhedron (3-polytope) P in ordinary three-dimensional space let S(P) denote its surface, that is, the union of its (closed) 2-faces. If P 1 and P 2 are given polyhedra, they are said to be…

### Areas of spherical polyhedral surfaces with regular faces

- Mathematics
- 2018

For a finite planar graph, it associates with some metric spaces, called (regular) spherical polyhedral surfaces, by replacing faces with regular spherical polygons in the unit sphere and gluing them…

### Small Icosahedral Clusters

- Art
- 2018

In geometry, any polyhedron with 12 faces is named a dodecahedron, among which only one is the regular dodecahedron (i.e. the Platonic solid), composed of 12 regular pentagonal faces, 3 of which…

### Convex polyhedra whose faces are equiangular or composed of such

- Mathematics
- 1978

Depending on the type, considering only the topological structure of the network of faces, and the angles of corresponding faces at corresponding vertices, convex polyhedra in R3, each face of which…

### Definitions in Polytopes

- Mathematics
- 2018

Multi-shell clusters represent complex structures, the study of which needs rigorous definitions in graph theory, geometry, set theory, etc. Within this chapter, main definitions for polyhedra,…

## References

SHOWING 1-10 OF 11 REFERENCES

### Generators and relations for discrete groups

- Mathematics
- 1957

1. Cyclic, Dicyclic and Metacyclic Groups.- 2. Systematic Enumeration of Cosets.- 3. Graphs, Maps and Cayley Diagrams.- 4. Abstract Crystallography.- 5. Hyperbolic Tessellations and Fundamental…

### The Thirteen Books of Euclid's Elements

- HistoryNature
- 1927

THE preface to the first edition of Sir Thomas Heath's translation of Euclid's “Elements” begins with the following quotation from De Morgan: “There never has been, and until we see it we never shall…

### Regular and semi-regular polytopes. II

- Mathematics
- 1940

Groupes de reflexion a 4 dimensions. Certains sous groupes d'indice petit. Construction de Wythoff et ses consequences numeriques. Polytopes a 4 dimensions. Nids d'abeilles a 4 dimensions. L'analogue…

### Mathematical Recreations and Essays

- MathematicsNature
- 1905

THIS edition differs from the third by containing chapters on the history of the mathematical tripos at Cambridge, Mersenne's numbers, and cryptography and ciphers, besides descriptions of some…

### Thirteen Books of Euclid's Elements

- Mathematics
- 1908

There is disclosed a telephone number indexing device which comprises a casing having an upper opening and housing therein a plurality of cards on which telephone numbers are recorded. When any card…

### Zalgaller and others, O praviVnogrannyh mnogogrannikah, Vestnik Leningrad

- Univ. Ser. Mat. Meh. Astron.,
- 1965

### Over een bewering van Euclides

- Simon Stevin
- 1947

### Convex polyhedra with regular faces (preliminary report), Abstract 576-157

- Notices Amer. Math. Soc,
- 1960