1984

Polygons and polyhedra

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See other Examples of Integrated, Multi-set Concept Schemes. The concept scheme described here is discussed in the paper on: Patterns of N-foldness: comparison of integrated multi-set concept schemes as forms of presentation (1984). This was prepared for a sub-project meeting of the Forms of Presentation group of the Goals, Processes and Indicators of Development (GPID) project of the United Nations University (UNU). The annexes were published in Patterns of Conceptual Integration. Brussels, UIA, 1984, pp. 161-204

Summary

The constraints governing these structures are those which govern order in space and may therefore be considered as significant for any investigation of order. The points are extracted from two books:

• Keith Critchlow. Order in Space; a design source book. Thames and Hudson, 1969
• Anthony Pugh. Polyhedra; a visual approach. University of California Press, 1976

3.1 "...the equilateral triangle, the square and the hexagon. These three are the only regular polygons that cover a plane surface, and are Known as the equipartitions of the plane surface. Earlier deductions have shown that there cannot be less than three polygons nor more than six around a vertex." (Critchlow, p. 60) "We find that the first two of the three regular equipart- itions. ..reciprocate each other, i.e. the dual of the triangular pattern is the hexagon and vice versa. The square is its own dual." (Critchlow, p. 76)

3.2 Three types of polygon are required to construct the Platonic polyhedra: triangle, square, pentagon (Pugh, p. 1)

4.1 There are only four facially regular prisms and antiprisms using the faces of the Platonic polyhedra (see 3.2): triangular prism, pentagonal prism, square antiprism, pentagonal antiprism (excluding the Platonic polyhedra: cube, tetrahedron, octahedron) (Pugh, 21) There four duals are: triangular dipyramid, pentagonal dipyramid, trapezoidal octahedron, trapezoidal deca- hedron. (Pugh, p. 44) (N.B. The dipyramids have isocèles triangles)

4.2 There are only four nonconvex facially regular polyhedra (Kepler-Poinsot polyhedra): small stellated dodecahedron, great stellated dodecahedron, great dodecahedron, great icosahedron. The dual of one Kepler Poinsot polyhedron is another Kepler Poinsot polyhhedron. (Pugh, pp. 84-7)

5.1 There are only five polyhedra composed of nonintersecting regular, plane, convex polygons with straight sides (the Platonic polyhedra): tetrahedron, octahedron, cube, icosahedron, (pentagonal) dodecahedron. (Pugh, p.2-3) The cube and octahedron are duals, as are the icosahedron and the dodecahedron. The tetrahedron is self-dual. (Pugh, p. 19)

5.2 There are only five convex deltahedra: triangular dipyramid, pentagonal dipyramid, 12-faced deltahedron, 14-faced deltahedron, 16-faced deltahedron excluding the Platonic polyhedra: tetrahedron, octahedron, icosahedron). (Pugh, p. 35) (N.B. The dipyramids have equilateral triangles)

6.1 Six types of polygon are required to construct the 5 Platonic and 13 Archimedean polyhedra: triangle, square, pentagon, hexagon,octagon, decagon (Pugh 26)

6.2 There can be no more than six polygons around the vertex of a mosaic, grid, lattice or tesselation (Critchlow, p 60)

6.3 There are six basic ways in which one polyhedron can be joined to another:

• - vertex to vertex
• - vertex of one figure pentetrates other
• - edge to edge until a common series of edges along which
• - face to face the figures can be joined is found
• - edge of one figure penetrates other figure until a suitable common plane at which the figures may be joined is found
• - face of one figure penetrates the other until a suitable common plane is found (Pugh, p.46)

7.1 There are7 facially regular prisms and antiprisms using the 3 polygonal faces of the Platonic polyhedre (see 3.2): triangular prism, square prism/cube, pentagonal prism, tetrahedron, triangular antiprism/octahedron, square antiprism, pentagonal antiprism (i.e. including three Platonic polyhedra) (Pugh, p.27)

8.1 A deltahedron is a polyhedron with faces which are all equilateral triangles. There is an infinite number of concave deltahedra...However, there are only eight convex deltahedra: tetrahedron, octahedron, icosahedron, triangular dipyramid, pentagonal dipyramid, 12-faced deltahedron, 14-faced deltahedron, 16-faced deltahedron. (Pugh, pp. 34-5)

8.2 There are 4 prisms and antiprisms (using the 3 Platonic faces) with their 4 duals (see 4.1)

8.3 There are 8 polyhedra which may be used in any combination as all-space filling solids, namely the eight deltahedra (see 8.1) (Critchlow, App. 2)

8.4 There are 8 polyhedra which are all-space filling on their own. individually: triangular prism, cubic prism/cube, hexagonal prism, truncated tetrahedron (produced), truncated octahedron, rhombic dodecahedron, twist rhombic dodecahedron, rhombex dodecahedron (Critchlow, App 2)

8.5 There are 8 groups of three polyhedra which are all-space filling:

• - cube, terahedron, rhornbicuboctahedron
• - cube, cuboctahedron, rhombicuboctahedron
• - cube, truncated octahedron, truncated cuboctahedron
• - truncated tetrahedron, cuboctahedron, truncated octahedron
• - truncated tetrahedron, truncated cube, truncated cuboctahedron
• - cube, triangular prism, hexagonal prism
• - cube, triangular prism, dodecahedral prism
• - cube, hexagonal prism, dodecahedral prism (Critchlow, App 2)

B.6 (Excluding 3.1 above) "Thus there remain only eight conditions of meeting, which it will be seen, give rise to twenty-two new patterns or grids (on a plane surface). These can be divided by symmetry into those whose vertices are similar on each occasion and those whose vertices vary. There are eight semi-regular equipartitions of the plane surface and fourteen demi-regular equipartitions." (Critchlow, p 60) "The semi-regular patterns...each have their characteristic dual." (Critchlow, p. 76)

10.1 There are 5 convex deltahedra (excluding the Platonic polyhedra) together with their duals (see 5.2)

10.2 There are 10 groups of two polyhedra which are all-space filling:

- octahedron, tetrahedron - triangular prism, cubic prism/cube - octahedron, cuboctahedron - triangular prism, hexagonal prism - octahedron, truncated cube - triangular prism, dodecahedral prism - tetrahedron, truncated tetrahedron - cube, edge truncated cube - cubic prism/cube, octagonal prism - truncated cuboctahedron, octag. prism (Critchlow. App 2)

10.3 There arel0 facially regular prisms and antiprisms using the 6 polygonal faces of the Platonic and Archimedean polyhedra (although excluding the latter): triangular prism, pentagonal prism, hexagonal prism, octagonal prism, decagonal prism, square antiprism, pentagonal antiprism, hexagonal prism, octagonal antiprism, decagonal entiprism (Pugh 27)

11.1 There are 3 regular and 8 semi-regular equipartitions of the plane surface (Critchlow. p 76)

12.1! There are 12 paired Archimedean polyhedra (see 13.1)

13.1 There are 13 Archimedean polyhedra in which similar arrangements of regular, convex polygons of two or more different kinds meet at each vertex of the polyhedron [which can be circumscribed by a tetrahedron, with 4 common faces):

- truncated octahedron - truncated icosahedron - cuboctahedron/vector equilibrium - icosidodecahedron - truncated cuboctahedron - truncated icosidodecahedron - snub cube - snub dodecahedron - (small) rhombicuboctahedron - (great) rhombicosidodecahedron - truncated cube - truncated dodecahedron - truncated tetrahedron (Critchlow, App 1) There are 13 duals to these polyhedra (Pugh, p. 43)

13.2 There are 13 facially regular prisms and antiprisms using the 6 polygonal faces of the Platonic and Archimedean polyhedra (namely 10.3 above, plus square prism/cube, tetrahedron, triangular antiprism/octahedron). (Pugh p.27)

14,1 "There are eight semi-regular equipartitions of the plane surface and fourteen demi-regular equipartitions" (Critchlow, p.60) [see 8.6 above) The 14 have only four new duals (Critchlow, p.76)

16.1 There are 8 semi-regular equipartitions of the plane surface, together with their 8 duals (see 8.6 above)

18.1 There are 5 Platonic polyhedra (see 5.1) and 13 Archimedean polyhedra (see

13.1), each of the latter being circumscribable by one of the former so that all of its vertices lie evenly arranged on the faces or edges of the circumscribing figure. (Pugh p.22)

19.1 There are 8 semi-regular equipartitions of the plane surface, together with their 8 duals; there are 3 regular equipartitions of the plane surface which are their own duals (see 3.1 and 16.1 above)

20.1 There are 10 facially regular prisms and antiprisms using the 6 polygonal faces of the Platonic and Archimedean polyhedra, together with 10 duals (see 10.3 above)

22.1 There are eight semi-regular equipartitions of the plane surface and fourteen demi-regular equipartitions (see 8.6 above)

23.1 Platonic polyhedra (5), Archimedean polyhedra (13), convex deltahedra (5)

23.2 Facially regular prisms and antiprisms (see 10,3) plus their 10 duals, plus the 3 Platonic polyhedra which can be included and which are their own duals (tetrahedron, cube, octahedron)

25.1 Regular (see 3.1), semi-regular (see 8.6)and demi-regular (see 14.13 equipartitions of the plane surface

26.1 Archimedean polyhedra (see 13.1) and their duals

28,1 Platonic, Archimedean polyhedra, together with facially regular prisms and antiprisms (see 10.3)

31.1 Archimedean polyhedra and their duals (see 26.1), plus the Platonic polyhedra which are their own duals

92.1 Convex polyhedra with regular faces (excluding Archimedean, Platonic, prisms and antiprisms, except as combinations) (Pugh, p.26)