1. University of Washington Seattle, WA 98195-4350 USA grunbaum@math.washington.edu Branko Grünbaum SMALL POLYHEDRAL MODELS OF THE TORUS, THE PROJECTIVE PLANE, AND THE KLEIN BOTTLE

2. For which integers n are there polyhedra that are topologically equivalent to a torus, with all n faces triangles ? Or all faces quadrangles, or pentagons,... ?

3. For which integers n are there polyhedra that are topologically equivalent to a torus, with all n faces triangles ? Or all faces quadrangles, or pentagons,... ? We need to stop at hexagons, since no torus can have all faces with seven or more sides.

4. For which integers n are there polyhedra that are topologically equivalent to a torus, with all n faces triangles ? Or all faces quadrangles, or pentagons,... ? We need to stop at hexagons, since no torus can have all faces with seven or more sides. Joint work with Lajos Szilassi

5. The question is: For which n can the torus be geometrically (i)triangulated (ii)quadrangulated (iii) quintangulated (iv)hexangulated with n faces ? The torus should be acoptic, that is, with simple faces and without selfintersections.

6. TRIANGULATIONS It is well known that combinatorial (topological) triangulations of the torus with n faces exist for all even n with n ≥ 2, and only for such n. Geometric triangulations exist if and only if n is even and n ≥ 14. n = 14: Császár torus

7. For triangulations with n ≥ 16: Start with a triangulation with n–2 faces and attach a tetrahedron on one of the faces for a net increase of two faces.

8. For triangulations with n ≥ 16: Start with a triangulation with n–2 faces and attach a tetrahedron on one of the faces for a net increase of two faces. There exist isogonal triangulations of the torus with n faces if and only if n = 4m, m ≥ 5.

9. OVERARCHING FACES: The intersection of two faces has more than one connected component.

10. A far-reaching generalization of these results is the following theorem of Archdeacon et al.: Every topological triangulation of the torus with no overarching faces is isomorphic to a geometric triangulation.

11. QUADRANGULATIONS Quadrangulations with n non-overarching faces exist for all n ≥ 9, except possibly for n = 10, 11.

12. QUADRANGULATIONS Quadrangulations with n non-overarching faces exist for all n ≥ 9, except possibly for n = 10, 11. Two basic constructions First construction: For all integers p ≥ 3 and q ≥ 3 we can construct "picture frames" for p-sided "pictures", with q-sided cross-sections. These give quadrangulations with n = p q, thus yielding the values n = 9, 12, 15, 16, 18, 20,....

13. A picture frame for a triangular picture with pentagonal cross section, and a frame for a pentagonal picture with a triangular cross section.

14. Second construction: To a face of a given quadrangulation attach a suitable image of a cube. This increases the number of faces by 4. From n = 9 we get n = 13 and then n = 17. Then the consecutive values n = 15, 16, 17, 18 are available, hence adding multiples of 4 yields all n ≥ 15.

15. The only still missing value is n = 14. An example is shown here:

16. If overarching faces are admitted, then quadrangulations are possible for all n > 9. Example with n = 10.

17. An example with n = 11 faces. Conjecture. There exist no geometric acoptic quadrangulations with n ≤ 8 quadrangles.

18. A far-reaching conjecture is: Every topological quadrangulation with no overarching faces can be realized geometrically. Only a few examples of such topological quadrangulations are known for which it has been proved that they cannot be realized by polyhedra with convex faces. They all seem to be realizable with simple faces that are not necessarily convex.

19. QUINTANGULATIONS For every even n ≥ 12 there are convex-faced quintangulations of the torus, except possibly for n = 14. Basic construction:

20. Simple variants of this construction yield quintangulations with p-fold rotational symmetry and with n = 2 p q faces, for all p ≥ 3, q ≥ 2. In particular, n = 12, 16, 18, 20, 24, 28,... are obtained. Attaching a copy of a dodecahedron yields an increase of 10 faces, thus establishing the claim. In the previous slide p = 7, q = 2.

21. CONJECTURE: There are no quintangulations with n convex faces where n = 14 or n ≤ 10. It is not clear what happens if one does not insist that the faces are convex.

22. HEXANGULATIONS Acoptic hexangulations with n faces, with no overarching faces, exist for n = 7, and for all n = p q with p ≥ 3, q ≥ 3. The case n = 7 is the well-known Szilassi polyhedron

23. Hexangulations with n = p q for p ≥ 3, q ≥ 3 can be constructed by starting with trapezohedra (Catalan polyhedra, that are polar to the antiprisms). For each p ≥ 3 such a polyhedron P has 2p quadrangular faces, p of which meet at each of two apices of P. By intersecting such a polyhedron P with a p-sided prism, having its axis coinciding with the axis of P and rotated appropriately, the resulting tunnel has p hexagonal sides, and all 2p sides of P become hexagons as well. Example: p = 3, q = 3 so n = 9

24. An example with p = 5, q = 3, so n = 15.

25. An example with p = 3, q = 5, so again n = 15.

26. Conjecture. The only acoptic hexangulations with n faces and no overarching that have rotational symmetry of order 3 or more are those with n = p q, with p, q ≥ 3. There are hexangulations lacking such symmetry for some other values of n. These values have not been characterized. If overarching faces are allowed, some other symmetric possibilities arise. Here is a hexangulation with 8 hexagons. It was found by J. Schwörbel.

27. In contrast to the situation concerning quintangulations, it is well known that no hexangulation of the torus can have only convex faces. Conjecture. Every hexangulation of the torus has at least six non-convex faces. Conjecture. Polyhedra of Kepler-Poinsot type admit hexangulations with n faces for all n ≥ 7.

28. Example. A hexangulation with n = 11 faces. It has two pairs of intersecting faces, and one selfintersecting face.

29. One may wonder how far all these constructions can be generalized. Long ago I proposed the following Conjecture. Every cell-complex decomposition, without overarching elements, of an orientable 2-manifold is realizable by an acoptic polyhedron. The theorem of Archdeacon et al. confirms this in a very special case, while a result of Bokowski et al. shows that it fails for triangulations of a 6-manifold. As far as I know, there is no information in the case overarching faces are admitted. No characterization of spherical acoptic polyhedra is known.

30. In the second part we shall be concerned with polyhedral realizations of non-orientable manifolds. By a basic topological result such polyhedra cannot be acoptic.

31. In this part we shall look mainly for polyhedra that realize or represent the projective plane, the Klein bottle, and the Möbius band. These are the simplest non-orientable 2-manifolds. This will lead to considerations of non-convex polyhedra, polyhedra with selfintersections, and most significantly, polyhedra with selfintersecting faces.

32. During most of the Twentieth Century polyhedra with selfintersections have been neglected, except for some very special classes. In particular, their topological properties have not been investigated. This is part of our goal here. Another part is to simply show some of these polyhedra, in order to illustrate their mathematical and esthetic aspects.

33. Selfintersecting polygons are of primary importance as faces of the polyhedra we consider. Here are some examples of such polygons, and the names I will use. Bow-tie (quadrangle)‏ Hexagram Pentagram In all cases we need to distinguish between the polygonal line and the polygon as a 2-dimensional part of the plane.

34. USUAL WAYS OF PRESENTING THE REAL PROJECTIVE PLANE P 2

35. USUAL WAYS OF PRESENTING THE REAL PROJECTIVE PLANE P 2 Euclidean plane E 2 together with points and line at infinity Family of lines and planes through the origin of E 3 Points and great circles of a sphere, with antipodal points identified Circular disk in E 2, with diametral points identified

36. Convex polyhedra with center, in which opposite elements (vertices, edges, faces) are identified. These are often called hemi-polyhedra. POLYHEDRAL MODELS OF THE PROJECTIVE PLANE Regular dodecahedron with opposite elements identified

37. Convex polyhedra with center, in which opposite elements (vertices, edges, faces) are identified. These are often called hemi-polyhedra. POLYHEDRAL MODELS OF THE PROJECTIVE PLANE Planar polyhedral maps, with boundary identification, are analogues of the circular disk model in E 2, with diametral points identified. 1 2 34 5 1 2 3 4 5 Regular dodecahedron with opposite elements identified The corresponding polyhedral map

38. CAN A POLYHEDRAL MAP OF THE PROJECTIVE PLANE P 2 BE REALIZED BY AN ACTUAL POLYHEDRON ?

39. This cannot be an acoptic (selfintersection-free) polyhedron, since acoptic polyhedra are orientable, while the projective plane is non-orientable. Allowing intersections of faces, can all faces be simple polygons? CAN A POLYHEDRAL MAP OF THE PROJECTIVE PLANE P 2 BE REALIZED BY AN ACTUAL POLYHEDRON ?

40. This cannot be an acoptic (selfintersection-free) polyhedron, since acoptic polyhedra are orientable, while the projective plane is non-orientable. Allowing intersections of faces, can all faces be simple polygons? CAN A POLYHEDRAL MAP OF THE PROJECTIVE PLANE P 2 BE REALIZED BY AN ACTUAL POLYHEDRON ? YES !  (hemi-cuboctahedron)‏

41. The projective plane P 2 is a 2-manifold, as is the polyhedral map with boundary identification. However, the heptahedron (or hemi- cuboctahedron) is not a manifold, since each neighborhood of each of its six vertices has an essential selfintersection. Vertex figure is a bow-tie quadrangle Such a singular point is known as a Whitney umbrella  …

42. If the heptahedron were a 2-manifold, which manifold would it be ? Every closed 2-manifold M is characterized by its orientability (orientable or non-orientable) and by its Euler characteristic  (M) = V – E + F.  (M) = 2 ==> M is the orientable 2-sphere.  (M) = 1 ==> M is the non-orientable projective plane.  (M) = 0 ==> M is either the orientable torus, or else the non-orientable Klein bottle.

43. If the heptahedron were a 2-manifold, which manifold would it be ? Every closed 2-manifold M is characterized by its orientability (orientable or non-orientable) and by its Euler characteristic  (M) = V – E + F. Since  (H) = 1, if H denotes the heptahedron, the only manifold it could be is the projective plane – but the Whitney umbrella is an impediment.  (M) = 2 ==> M is the orientable 2-sphere.  (M) = 1 ==> M is the non-orientable projective plane.  (M) = 0 ==> M is either the orientable torus, or else the non-orientable Klein bottle.

44. There are many possibilities for realizations of hemi-polyhedra (that is, the projective plane) and not all have Whitney umbrellas. Hemi- dodecahedron Reinhardt 1885

45. Hemi-dodeca- hedron (L. Szilassi, 2007)‏ Conjecture (Szilassi): Every realization of the hemi-dodecahedron has at least one selfintersecting face. 1 2 34 5 10 9 8 7 6 9 8 7 6 Face [1,2,3,4,5] is selfintersecting, faces [1,8,9,10,2], [3,2,10,6,7], [4,5,6,10,9], [1,5,6,7,8] and [3,7,8,9,4] are simple polygons; the last two intersect.

46. Problem: The classification of 2-manifolds implies that every manifold is homeomorphic to one of the standard representatives. For  (M) ≥ 0 this means that M is homeomorphic to one of the four we have listed three slides back. But just as a Whitney umbrella cannot be homeomorphic to a disk, neither can a selfintersecting polygon. ?

47. The problem has nothing to do with non-orientability, or the projective plane –– it is inherent in selfintersection of faces. The prism with bow-tie basis is isomorphic to the cube, but is it homeomorphic ? 1 2 3 4 6 5 7 8 1 2 3 4 6 5 7 8

48. The regular dodecahedron and the great stellated dodecahedron are isomorphic, as shown by the labels. But are they homeomorphic? That is, is there is a 1-to-1 map that is continuous and has a continuous inverse. 13 52 34 21 45 24 35 41 12 54 23 15 51 43 25 31 53 32 14 42 12 45 23 51 34 25 13 42 41 53 14 35 24 31 52 54 32 15 21 43

49. Dilemma: Do we stop thinking of polyhedra with selfintersecting faces as manifolds (as topology does, for at least the last century), or do we find a way to interpret selfintersecting polygons as homeomorphic to simple polygons ? The regular dodecahedron and the great stellated dodecahedron are isomorphic, as shown by the labels. But are they homeomorphic? That is, is there is a 1-to-1 map that is continuous and has a continuous inverse. 13 52 34 21 45 24 35 41 12 54 23 15 51 43 25 31 53 32 14 42 12 45 23 51 34 25 13 42 41 53 14 35 24 31 52 54 32 15 21 43

50. A similar problem arose, and was solved, long ago in connection with curves and polygonal lines. It is obvious that as a set of points, the bow-tie quadrangular line is not homeomorphic to the boundary of the square (or to a circle).

51. To establish a homeomorphism, the concept of immersion was accepted as the solution: A crossing-point (such as x) is considered as representing two distinct points, one on each branch of the curve or polygonal line. This makes possible a parametrization of the selfintersecting curve or polygonal line.

52. To establish a homeomorphism, the concept of immersion was accepted as the solution: A crossing-point (such as x) is considered as representing two distinct points, one on each branch of the curve or polygonal line. This makes possible a parametrization of the selfintersecting curve or polygonal line. With this understanding, and obvious generalizations, selfintersection points of polygonal lines (or curves) do not prevent homeomorphisms of the parametrized line with simple polygonal lines, or with a circle.

53. To establish a homeomorphism, the concept of immersion was accepted as the solution: A crossing-point (such as x) is considered as representing two distinct points, one on each branch of the curve or polygonal line. This makes possible a parametrization of the selfintersecting curve or polygonal line. With this understanding, and obvious generalizations, selfintersection points of polygonal lines (or curves) do not prevent homeomorphisms of the parametrized line with simple polygonal lines, or with a circle. Note: we are dealing with a homeomorphism of the parametrized lines or curves –– not with a homeomorphism of the plane.

54. The concept of immersion and the appropriate parametrization was long ago extended to cover segments of multiple points, as well as whole regions of such points. An example was provided by Reinhardt, in comments to the 1886 edition of Möbius’ collected works. His drawing of the Möbius band (as proposed by Möbius) is shown at left, while a version of the band, and of the hemi-icosahedron obtained by constructing a pyramid over the band, are shown in the other parts. These are taken from a book by Apéry (1987).

55. We need to validate the use of the Euler characteristic in deciding which manifold is represented by a polyhedron that may contain selfintersecting faces. By relying on the immersion and parametrization concepts, we know that a simple polygonal line is homeomorphic with the polygonal line that determines a face, even if the polygonal line is selfintersecting. Now we need to extend this homeomorphism in a way that the interior of the simple polygon maps onto a kind of “interior” of the selfintersecting polygon. Where is the quadrangular “interior” of the bow-tie that is homeomorphic to the interior of the square?

56. We need a different kind of immersion. To avoid misunder- standings, we call it infolding. In the infolding of the bow-tie quadrangle, or of the selfintersecting pentagon in the Szilassi pentagon, we need each crossing point to represent a segment of points. How such a parametrization can be done in a consistent and bicontinuous way is illustrated in the next slide. It shows an isotopy (continuous deformation) from a parametrized bow-tie to a square. The point parameters throughout the isotopy are (x, c), the isotopy parameter is t, and the points have coordinates (x, y). All parameters go from –1 to +1. The coordinate y is given by y = ((t – 1)x + t + 1)c/2

57. (1,–t)‏ (1,t)‏ (-1,1)‏ (-1,-1)‏ (-1,c)‏ (1,t c)‏ t = –1 (1,–t)‏ (1,t)‏ (-1,1)‏ (-1,-1)‏ (-1,c)‏ (1,t c)‏ t = –0.5 (1,t)‏ (-1,1)‏ (-1,-1)‏ (-1,c)‏ (1,t c)‏ t = 0 (1,t)‏ (1,–t)‏ (-1,1)‏ (-1,-1)‏ (-1,c)‏ (1,t c)‏ t = 0.5 (-1,1)‏(1,1)‏ (-1,-1)‏(1,–1)‏ (-1,c)‏ (1,c)‏ t = 1 y = ((t – 1)x + t + 1)c/2

58. For a different variety of infolding, consider the regular dodecahedron, and the great stellated dodecahedron. They are isomorphic, so the great stellated dodecahedron should be homeomorphic to a sphere –– even though it is well known that the density at its center is 3. 13 52 34 21 45 24 35 41 12 54 23 15 51 43 25 31 53 32 14 42 12 45 23 51 34 25 13 42 41 53 14 35 24 31 52 54 32 15 21 43

59. Here an isotopy connects a pentagram to a bow-tie quadrangle, which is isotopic to a square, which is isotopic to a pentagon. Thus the pentagram is homeomorphic to the pentagon, hence the great stellated dodecahedron is indeed homeomorphic to the Platonic dodecahedron.

60. It can be shown that the infolding approach generalizes to all polyhedra, and that therefore the criteria of orientability and Euler characteristic are sufficient to establish each polyhedron (in suitably parametrized form) as homeomorphic to a (compact) 2-manifold. All it takes is to interpret each selfintersection point as representing a whole segment in a suitable parametrization. A simple way to formulate this conclusion is: For suitably parametrized polyhedra ISOMORPHIC => HOMEOMORPHIC Here are a few examples of the application of this result.

61. Examples of connels (= conical tunnels) formed by bow-ties. Like tunnels, they can be used to replace pairs of faces that are related by symmetry in a point. If the pair was part of an orientable polyhedron, the resulting polyhedron is orientable as well. Each replacement reduces the Euler characteristic by 2. A simple example is shown in the next slide.

62. Torus: 4 squares, one connel with 4 bow-tie faces, 16 edges, 8 vertices,  = 0, orientable Faces: ABFE, BCGF, CDHG, DAEH, ADFG, DCEF, CBHE, BAGH ABBCCDDA EFFGGHHE ABBCCDDA EFFG GHHE A B C D E F G H A B C D E G H F

63. A selfdual torus, isomorphic to the preceding one: 4 rectangles, 4 bow-tie faces, 16 edges, 8 vertices (there are overarching faces). Described by K. Merz (1935)‏ This torus, and the one in the preceding slide, have fewer edges than the minimum number established by U. Brehm (1990) for a class of “polyhedral maps” more restricted than allowed here.

64. 01 2 3 4 024130 130241 A selfdual torus consisting of five bow-tie quadrangles, with five vertices and ten edges. All faces are in one plane. Some pairs of faces are overarching (have intersection with more than one component).

65. A projective plane consisting of five coplanar convex quadrangles (only one is shown filled in, for easier visualization). Some pairs overlap on triangular regions. This is the hemi-5-sided-trapezohedron (dual of antiprism). The center is a singular point (vertex figure has rotation number 2). Analogous projective planes exist for all odd n ≥ 5.

66. Projective plane Modified from an example of F. Apéry (1987)‏ 6 vertices, 14 edges, 9 faces (8 triangles, one bow-tie)‏

67. Projective plane Taking two copies with coinciding bow-ties, and eliminating the bow-ties yields an ORIENTABLE polyhedron. This is a hemi-polyhedron of a torus, which is itself a torus. 6 vertices, 14 edges, 9 faces (8 triangles, one bow-tie)‏

68. 6 quadrangles (3 convex, 3 bow-tie), 12 edges, 7 vertices,  = 1 Projective plane, hemi-rhombic dodecahedron This model has fewer edges than the minimum for a more restricted kind of “polyhedral maps” considered by U. Brehm (1990)‏

69. Projective plane: Heptahedron, dual to the hexahedron of the previous slide. 3 bow-tie quadrangles, 4 triangles, 12 edges, 6 vertices Two copies joined at bases give a Klein bottle, with 6 bow-tie quadrangles, 6 triangles, 21 edges, 9 vertices

70. Projective plane: 6 quadrangles, 1 hexagon, 15 edges, 9 vertices Put two copies together, and eliminate hexagons: Klein bottle with 12 quadrangles, 24 edges, 12 vertices

71. Faces: ABCD, BAEF, FEHG, GHDC, AEDH, ADEH, BCFG, BFCD D G HE FF F C GGF C CCBB BB A AA ADD E E E HHD H G Klein bottle: 8 faces, 16 edges, 8 vertices,  = 0, non-orientable

72. Klein bottle: 12 quadrangles, 20 edges, 8 vertices, non-orientable no overarching faces 1 2 3 4 5 6 7 8 9 10 11 12 1 2 3 4 5 6 7 8 9 10 11 12 1 2 3 4 5 6 7 8 9 10 11 12

73. Klein bottle: 12 quadrangles8 quadrangles, 2 hexagons 1 2 3 4 5 6 7 8 9 10 11 12 1 2 3 4 5 6 7 8 9 10 11 12 1 2 3 4 5 6 7 8 9 10 11 12 1 2 3 4 5 6 7 8 9 10 11 12

74. Six triangles of the mantle (only one shown at right), a connel (conical tunnel) of three bow-ties (two shown at left), 15 edges, six vertices. Orientable. = 0. Torus with overarching faces.

75. Two triangles, six bow-ties, 15 edges, six vertices,  = –1. Overarching faces. Non-orientable, of genus 3

76. Various interesting polyhedra are possible with more complicated selfintersecting faces. A few examples are shown next. For simplicity, all are isogonal, that is, have vertices that are equivalent under symmetries of the polyhedron.

77. (3.6*.3 #.6*)‏ An isogonal faceting of the truncated tetrahedron. This is topologically a torus, in which each vertex is incident with one small triangle, one large triangle and two hexagrams.

78. (3.6*.3 #.6*)‏ An isogonal faceting of the truncated tetrahedron. This is topologically a torus, in which each vertex is incident with one small triangle, one large triangle and two hexagrams. The same type of polyhedron is possible for all hexagrams in the isotopy (metamorphosis) shown.

79. 2 1 34 5 61,6 2,34,5 1 2 34 5 61 2 3 4 5 6 1 2 34 5 6 1 2 34 5 6 {6} 1,4 2,5 3,6 {6/2} A view of the metamorphosis

80. An isogonal faceting of the truncated dodecahedron involving decagrammatical faces. Each vertex meets one triangle, one regular pentagon, and two decagrams, as shown at left. The resulting polyhedron is shown at right without the pentagons, to make it more intelligible. (3.10*.5.10*)‏

81. Pentagons not shown. As in the case with hexagons we have seen earlier, there is a metamorphosis of decagons that connects two regular decagons through a continuum of isogonal ones. All are suitable to yield polyhedra of the same type.

82. Another isogonal faceting of the truncated dodecahedron involving decagrammatical faces. At each vertex one regular pentagon meets two decagrams. The resulting polyhedron is shown without the pentagons, to make it more intelligible. Orientable, 24 faces, 90 edges, 60 vertices, genus 4. (5.10*.10*)‏

83. (3.10*.10*.10*.10*)‏ An isogonal faceting of the truncated dodecahedron in which each vertex is incident with one triangle and four decagrams. Orientable, 44 faces, 150 edges, 60 vertices, genus 24.

84. Isogonal, one triangle and four bow-ties at each vertex. 60 vertices, 150 edges, 80 faces,  = –10. Non-orientable, genus 12.

85.  ö   The boundary of a Möbius band consist of a single closed curve. A realization of the Möbius band in which this curve is planar and bounds a topological disk is often called a cross-cap. Any cross-cap must have selfintersections. Here are two familiar versions.

86.             Adapted from Apéry  ö  

87.    Each has two Whitney umbrellas. The homeomorphism to the Möbius band needs justification. This is possible, but is rarely given.

88.    Each has two Whitney umbrellas. The homeomorphism to the Möbius band needs justification. This is possible, but is rarely given. What is the importance of cross-caps? They are needed in the basic theorem on the classification of closed 2-manifolds,

89. Orientable 2-manifold M Euler characteristic  (M) = 2 – 2g, g = (orientable) genus of M = (2 –  )/2 Non-orientable 2-manifold M Euler characteristic  (M) = 2 – g g = (non-orientable) genus of M = 2 - 

90. Orientable 2-manifold M Euler characteristic  (M) = 2 – 2g, g = (orientable) genus of M = (2 –  )/2 Non-orientable 2-manifold M Euler characteristic  (M) = 2 – g g = (non-orientable) genus of M = 2 -  Sphere: Orientable, g = 0,  = 2. Each orientable M of genus g ≥ 0 is homeomorphic to sphere with g handles. Each non-orientable M of genus g ≥ 0 is homeomorphic to sphere with g cross-caps. The classification of 2-manifolds:

91. A B C D E F  ö   Polyhedral Möbius band that is a cross-cap A B C F E D

92. A B C D E FA B C D E F A B C F E D A B C F E D A B C D E F A B C F E D = Transition from the traditional Möbius band to the polyhedral three-bow-ties cross-cap.

93. A variety of cross-caps Each is homeomorphic to the Möbius band by infolding.

94. The three bow-tie quadrangles AA*BB*, BB*CC* and CC*AA* overlap in the triangular region shown in white. The three bow-ties form a cross-cap, just as in the case their crossover points coincide. Another kind of cross-caps A B C B* A* C*

95. Despite the work of Brückner and others a century or more ago (or possibly because of it !?), the study of polyhedra with selfintersections and with selfintersecting faces has been neglected during much of the last 100 years. I hope the examples of such polyhedra and their applications to the topology of manifolds discussed here will lead to an awakening of interest in their study.