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Duality of Polyhedra

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Title: Duality of Polyhedra
Language: English
Authors: Gailiunas, P., Sharp, J.
Source: International Journal of Mathematical Education in Science & Technology. Sep 2005 36(6):617-642.
Availability: Customer Services for Taylor & Francis Group Journals, 325 Chestnut Street, Suite 800, Philadelphia, PA 19106. Tel: 800-354-1420 (Toll Free); Fax: 215-625-8914.
Peer Reviewed: Y
Page Count: 26
Publication Date: 2005
Document Type: Journal Articles
Numerical/Quantitative Data
Reports - Descriptive
Descriptors: Logical Thinking, Computer Graphics, Computer Simulation, Thinking Skills, Geometric Concepts, Geometry, Mathematics Instruction, Computation, Numbers
ISSN: 0020-739X
Abstract: Everyone is familiar with the concept that the cube and octahedron, dodecahedron and icosahedron are dual pairs, with the tetrahedron being self-dual. On the face of it, the concept seems straightforward; however, in all but the most symmetrical cases it is far from clear. By using the computer and three-dimensional graphics programs, it is possible to clarify the concept and explore new ideas. Moreover, it is an ideal topic for teaching clear logical thinking. (Contains 21 figures.)
Abstractor: Author
Number of References: 18
Entry Date: 2006
Access URL: https://taylorandfrancis.metapress.com/link.asp?target=contribution&id=W2M4840P05P54146
Accession Number: EJ729392
Database: ERIC
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  Value: <anid>AN0018528183;imt15sep.05;2019Feb27.14:39;v2.2.500</anid> <title id="AN0018528183-1">Duality of polyhedra. </title> <sbt id="AN0018528183-2">1. Introduction</sbt> <p>Everyone is familiar with the concept that the cube and octahedron, dodecahedron and icosahedron are dual pairs, with the tetrahedron being self-dual. On the face of it, the concept seems straightforward; however, in all but the most symmetrical cases it is far from clear. By using the computer and three-dimensional graphics programs, it is possible to clarify the concept and explore new ideas. Moreover, it is an ideal topic for teaching clear logical thinking.</p> <p>Duality is usually taught initially as a property of the Platonic solids, by pointing out that that the number of vertices of the cube and the number of faces of the octahedron are equal and vice versa. Similarly, the dodecahedron and icosahedron are dual pairs and the tetrahedron is self-dual. It is common to see drawings of this with, for example, a regular octahedron with its vertices sitting at the centre of a cube. This works for the regular polyhedra because of their symmetry, and is often cited as a way of obtaining the dual; for example Holden [<reflink idref="bib1" id="ref1">1</reflink>], uses it as a means of introducing duality. However, even the duals of the Archimedean solids (the Catalan solids) cannot be constructed by joining the centroids of the faces, but you <emph>can</emph> join the points where the faces of a Catalan solid touch the insphere to get its metric dual (the original Archimedean). To be fair to Holden, he does not discuss the joining of the centre of faces when describing the Archimedeans and their duals, but he does not make it explicit in his first definition that his description is a special case. This is a typical way that the half truth is perpetuated and why we have provided detailed examples, many of them non-symmetrical.</p> <p>In discussing why this is the case, and looking through the literature, we have had difficulty in finding a clear exposition of the subject. Even the relatively recent book on polyhedra by Cromwell [<reflink idref="bib2" id="ref2">2</reflink>] shies away from considering duality, which seems most unusual in such a comprehensive book, but is indicative of the lack of suitable source material. We were surprised that despite the idea being quite simple, discussion of the finer detail is often confused. There is the further complication that in some papers, the original definition is used in a way that does not follow the mathematical convention, although the subject matter is perfectly valid in its own right (for example de Villiers [<reflink idref="bib3" id="ref3">3</reflink>] has generalized van Aubel's theorem using a non- standard concept of duality). Because they are not conventional, we will not cover them but stick to a discussion of duality which is precise in defining terms and in using concepts in order to avoid confusion, and possibly errors. Another potential source of confusion is illustrated by Bakos [<reflink idref="bib4" id="ref4">4</reflink>], where he swaps the words cube and octahedron in discussing some dual configurations, without going into the finer detail of what he is actually discussing. So he claims that finding a configuration of a cube inscribed in an octahedron, with its vertices on the edges of the octahedron, is the dual problem of finding a configuration of an octahedron inscribed in a cube, with the octahedral vertices on the edges of the cube. Grünbaum and Shephard [<reflink idref="bib5" id="ref5">5</reflink>] discuss some aspects of these difficulties. However, in their resolution to demonstrate the extent of the muddle, they have potentially added to it, in some instances by failing to make clear which meaning they intend, and also by giving the impression that they accept certain polyhedra to be duals, based on a mistaken assumption. We will deal with these cases in detail later.</p> <p>This description of the duality of polyhedra has been made much easier by the use of computers to produce objects we can see and manipulate in three dimensions on the computer screen. Apart from the visual attraction of this as a teaching tool, the use of technology is an aid to the use of clear and logical thinking. We had to produce objects dual to the ones we were studying, so we had to be clear about what we were trying to achieve, in order to construct an algorithm. This points out another lesson that having to work from scratch is a better aid to learning than pressing buttons and entering data in someone else's program.</p> <hd id="AN0018528183-3">2. Polyhedra</hd> <p>Coxeter [<reflink idref="bib6" id="ref6">6</reflink>] defines a polyhedron as 'a finite, connected set of plane polygons, such that every side of each polygon belongs also to just one other polygon'. He excludes anomalies like pairs of pyramids with a common apex by including that the polygons surrounding each vertex form a single circuit. The polygons are <emph>faces</emph> of the polyhedron, and to add clarity we will distinguish their <emph>sides</emph> from <emph>edges</emph> of the polyhedron, since an edge corresponds with sides of two polygons. Similarly <emph>corners</emph> of polygons are distinguished from <emph>vertices</emph> of the polyhedron.</p> <p>In considering duals of some polyhedra we will need to include examples where the faces intersect one another, and where the faces may also have sides that intersect. This needs much wider definitions of polygons and polyhedra. Grünbaum [<reflink idref="bib7" id="ref7">7</reflink>] has developed suitable rigorous definitions by treating polygons as circuits, so that a convex polygon is a circuit where each corner is visited once, and none of the sides cross, while a star polygon has sides crossing. Polyhedra are constructed by joining such polygons along their sides, and Grünbaum gives many examples of the range of polyhedra allowable under these definitions. We will also need to consider polygons having sides of zero length, so that different corners are not necessarily at distinct points, and polygons that are actually different can look the same. A hexagon can look like a triangle, for example, if three of its sides are of zero length.</p> <p>We will also need to consider the planes that contain the faces, the (infinite) lines that contain the edges, and the points that contain the vertices of polyhedra. A point can have none, one or more vertices.</p> <hd id="AN0018528183-4">3. Types of duality</hd> <p>The term duality is used in a range of contexts, and its meaning varies, so we need to be clear about which one is being discussed, since a statement can be valid if one definition is used and invalid in another case. Many problems arise by using the word duality in an imprecise sense and in some cases by using the term in more than one sense without informing the reader that the sense has changed.</p> <hd id="AN0018528183-5">3.1. Combinatorial duality</hd> <p>A simple instance of duality occurs in the theory of planar graphs. Duality of graphs is usually defined in set theoretic terms, but there is a more intuitive construction. Given a connected graph consisting of a set of vertices connected by arcs that divide the plane into several regions, a vertex of the dual is identified with each region of the original graph, and arcs of the dual constructed between these vertices if, and only if, the corresponding regions have a common arc as a boundary. This gives a one to one correspondence between arcs of the graph and its dual, between regions of the graph and vertices of the dual, and between vertices of the graph and regions of the dual. This determines the numbers of elements (vertices, edges and so on) and their connectivity, but not other geometric properties, e.g. convexity.</p> <p>This idea can be applied in a very natural way to polyhedra that are simply connected and have no intersecting faces, since they can be mapped onto a planar graph (the Schlegel diagram), but it cannot apply to more general cases, for example, those having faces that are star polygons, since such polygons are non-planar. A natural extension, however, considers a pair of polyhedra to be combinatorial duals if there is a one to one correspondence between the faces of one and the vertices of the other such that faces that share an edge in one correspond to vertices that are joined by an edge in the other.</p> <p>A simplistic attempt deriving from these ideas leads to the supposed duals constructed by identifying the centres of faces with dual vertices. This will produce forms of the accepted duals of the Platonic polyhedra, since each has an insphere that touches its faces at their centres, but fails even with the Archimedeans, when the face centres may not be coplanar.</p> <hd id="AN0018528183-6">3.2. Projective duality</hd> <p>The axioms of projective geometry in the plane that two points are incident with just one line and that two lines are incident with just one point imply the principle of duality:</p> <p>Every definition remains significant and every theorem remains true by virtue of its dual theorem when the two pairs of concepts <emph>point/line</emph> and <emph>join/intersect</emph> are exchanged.</p> <p>The principle of duality in the plane is properly attributed to Gergonne (1826). Its extension to three-dimensional space and placing on analytical footing is due to Möbius [<reflink idref="bib8" id="ref8">8</reflink>].</p> <p>Employing the principle to obtain a dual theorem, or description we interchange according to a wider 'dictionary' than the one above. In the plane, the dictionary includes the following, with an item from either column being replaced by the corresponding one from the other</p> <p></p> <p> <ephtml> <table><tbody valign="top"><tr><td>point</td><td>line</td></tr><tr><td>passing through</td><td>lying on</td></tr><tr><td>concurrent</td><td>collinear</td></tr><tr><td>join</td><td>intersection</td></tr><tr><td>point on a curve</td><td>tangent to a curve</td></tr><tr><td>range (all the points in a line)</td><td>pencil (all the lines in a point)</td></tr><tr><td>locus</td><td>envelope</td></tr></tbody></table> </ephtml> </p> <p>When we move away from the plane some items in the dictionary change, for example with three-dimensional space we have:</p> <p></p> <p> <ephtml> <table><tbody valign="top"><tr><td>point</td><td>plane</td></tr><tr><td>Line</td><td>line</td></tr><tr><td>passing through</td><td>lying on</td></tr><tr><td>coplanar points</td><td>concurrent planes</td></tr><tr><td>coaxial planes</td><td>collinear points</td></tr><tr><td>join</td><td>intersection</td></tr><tr><td>range (all the points in a line)</td><td>pencil (all the planes in a line)</td></tr></tbody></table> </ephtml> </p> <p>Points and planes are now duals, and lines are dual to lines. As with combinatorial duality, we are only establishing a correspondence, albeit a more rigid one. In combinatorial duality, lines are topological 'rubber' line segments. In projective duality, lines are infinite straight lines, what would have been called right lines two hundred years ago to distinguish them from curved lines. In both cases we can <emph>create</emph> figures that are duals, but in neither case can we <emph>construct</emph> or <emph>calculate</emph> one from the other.</p> <hd id="AN0018528183-7">3.3. Reciprocity or polarity in a conic or quadric</hd> <p>In projective geometry, it is possible to <emph>construct</emph> specific figures, which retain the above 'dictionary' properties, by making use of the pole and polar construction with respect to a conic in the plane or quadric surface in space. This is really a fortuitous coincidence, but as Pedoe [<reflink idref="bib9" id="ref9">9</reflink>] in a paper entitled 'Notes on the history of geometrical ideas: II. The principle of duality', describes, it caused a great deal of confusion in the nineteenth century. When Gergonne described the principle of duality, Poncelet (one of the giants of geometry who were responsible for reviving geometry after it had been subjugated by Cartesian geometry) protested that it was nothing more than his method of reciprocation (polarity) with respect to a conic. The debate became very acrimonious with Poncelet devoting large parts of his books to denouncing all the major mathematicians for stealing his ideas.</p> <p>The confusion between duality and reciprocation still exists in many books and is part of the cause of the problems we are trying to resolve. For a discussion of the difference see Coxeter [<reflink idref="bib10" id="ref10">10</reflink>], quoted by Pedoe as concluding that, 'Since the general Pappus configuration is self-dual without being self-polar, the old controversy between Poncelet and Gergonne is settled in the latter's favour.' For our purposes the important point is that reciprocity is a more stringent requirement than duality, and figures that are reciprocals are necessarily dual.</p> <p>Pole and polar are terms which fit in the dictionaries above. In the plane the relation is between a polar point and a polar line, with respect to a conic. In three-dimensional space it is between a polar point and polar plane with respect to a quadric surface. The important point to realise is that in each case there is a one to one mapping of the points and lines in the plane, and of the points and planes of three dimensional space. It is not difficult to construct the polar line of a point with respect to a conic if the point is outside the conic and it is possible to construct tangents to it: draw two tangents to the conic from the point and join the points of tangency. The construction of the pole is as easy if the line cuts the conic in two real points: intersect the tangents at those points. For the general case (figure 1), the double column method which was popular in projective geometry books shows both the power of the concept and its elegance:</p> <p></p> <p> <ephtml> <table><tbody valign="top"><tr><td><bold>To find the polar line of point X</bold></td><td><bold>To find the pole of line x</bold></td></tr><tr><td>1. Choose any two lines (a and b) through X which intersect the conic in real points.</td><td>1. Choose any two points A and B on line x which have real tangents with the conic.</td></tr><tr><td>2. Intersect these lines and the conic in points P, Q (for a) and R, S (for b).</td><td>2. Draw the pair of tangents from each point, p, q (from A) and r, s (from B).</td></tr><tr><td>3. Draw the tangents p, q, r, s at P, Q, R and S.</td><td>3. Draw the tangent points of these lines P, Q, R and S for p, q, r, s.</td></tr><tr><td>4. Intersect tangents p, q in A and r, s in B.</td><td>4. Join points P, Q in line a and R, S in line b.</td></tr><tr><td>5. Join A and B to give the polar line x.</td><td>5. Intersect a and b to give the pole X.</td></tr></tbody></table> </ephtml> </p> <p>Graph: Figure 1. Pole and polar.</p> <p>Note how figure 1 is a self-dual diagram; it applies to both constructions. Consequently, it is easy to see that the construction is reversible, for example if you construct the polar line of a point with respect to a conic and then construct the pole of a line with respect to the same conic, you will arrive back at the original point.</p> <p>This is fine for constructing pole and polar in the plane, and although a similar construction can be applied in space, drawing in space is not as easy. The construction of the reciprocal of a polyhedron in general becomes quite involved, so the normal convention uses a sphere as the reciprocating quadric. This has the benefit that the line joining the centre of the sphere and the pole is perpendicular to the polar plane.</p> <p>The symmetry of the sphere also makes it easier to visualize the result of reciprocation, but applying it to the Platonic solids, and extrapolating the results is a root of some of the problems. In the case of Platonic polyhedra the sphere can be chosen to be the insphere, so that the vertices of the dual lie at the centroids of the faces; the circumsphere, so that the faces of the dual lie in the tangent planes at the vertices; or the sphere touching the mid-points of the edges, so that the edges of the dual pair mutually intersect at right angles. These relations are a consequence of the high symmetry of the Platonic solids, and it should not be assumed that they extend to polyhedra in general.</p> <p>Duals of symmetrical polyhedra have been widely studied, usually understood to be reciprocals with respect to a sphere. Wenninger [<reflink idref="bib11" id="ref11">11</reflink>], for instance, has given detailed directions for building models of such duals of uniform polyhedra.</p> <p>Even though reciprocation would seem to define an essentially unique dual the situation is not so clear in less symmetrical cases, when, in the absence of a centre of symmetry, there may not be an obvious centre for the sphere. Using a different centre (or indeed a quadric different from a sphere) produces a polyhedron that has a very different appearance. Changing the radius of the sphere simply gives a similar reciprocal, differing only in scale.</p> <hd id="AN0018528183-8">4. Misleading cases</hd> <p>We can now return to the examples in the introduction.</p> <hd id="AN0018528183-9">4.1. Joining the centres of the faces of a polyhedron gives a dual</hd> <p>It is certainly the case that joining the centroids of the faces of the Platonic solids gives the metrically regular reciprocal dual. For example joining the centres of faces of a cube gives a regular octahedron, and the insphere of the cube becomes the circumsphere of the octahedron.</p> <p>Going a stage further to the Archimedean solids no longer gives the correct accepted dual (the appropriate Catalan polyhedron). This is because the faces of the Archimedean solids have no common insphere. In fact the centroids of the faces of an Archimedean polyhedron having a common vertex are not in general coplanar, so the construction does not even give a polyhedron. Reciprocation will always ensure that concurrent planes give rise to coplanar points in the dual, so this problem will never arise if it is used. In fact this is the only mapping between points and planes that can achieve this, because any such mapping could then be followed by a reciprocation, so that concurrent planes would have been mapped to coplanar points, and then back to concurrent planes. Similarly coplanar points would be mapped to concurrent planes, then back to coplanar points. The situation with collinear points/planes would be similar, so that the combined mapping would preserve incidences, and be a projective transformation. This means that the original mapping is equivalent to a projective transformation, followed by the inverse of a reciprocation (which is also reciprocation). But this combination would simply be another reciprocation.</p> <p>The natural requirement that coplanar points are dual to concurrent planes has an important consequence when applied to polyhedra having coplanar faces, for example figure 2. Clearly all the vertices of coplanar faces are themselves coplanar, so the corresponding dual faces are concurrent at a point that is dual to the common plane. This means that the vertices dual to the coplanar faces are all at the same point, although we need to distinguish them since duality is a one to one mapping between faces and vertices.</p> <p>Graph: Figure 2. Reciprocation with coplanar faces.</p> <p>In figure 2(a), the two triangles in the same plane at the left become two vertices in the same point at the lower left of figure 2(b).</p> <p>Although duals of the Archimedeans cannot be constructed by joining the centroids of the faces, because their vertices lie on a sphere each Catalan solid has an insphere, and you <emph>can</emph> join the points where the faces of a Catalan solid touch the insphere to get its metric dual (the original Archimedean).</p> <hd id="AN0018528183-10">4.2. The Bakos configurations</hd> <p>By calling them solutions to dual problems Bakos [<reflink idref="bib4" id="ref12">4</reflink>] gives the impression that the following two compound polyhedra are a pair of duals. The first is a regular octahedron inscribed in a cube so that its vertices are one on each of six edges of the cube. The second consists of a cube inscribed in a regular octahedron so that its vertices are one on each of eight edges of the octahedron.</p> <p>Analysing through the logic of transliteration according to the projective dictionary for his first figure using the 'two-column' method yields the following result:</p> <p></p> <p> <ephtml> <table><tbody valign="top"><tr><td><bold /><bold>Figure 3(a)</bold><bold /></td><td><bold>Projective dual</bold></td></tr><tr><td>1. Each vertex of the octahedron lies on the edge of a cube.</td><td>1. Each face of the cube must lie on the edge of an octahedron.</td></tr><tr><td>2. Each face of the cube has one edge of the octahedron in common.</td><td>2. Each vertex of the octahedron must lie on one edge of the cube.</td></tr><tr><td>3. None of the vertices of the cube lie on edges of the octahedron.</td><td>3. None of the faces of the octahedron must lie on edges of the cube.</td></tr></tbody></table> </ephtml> </p> <p>Graph: Figure 3. Bakos configurations.</p> <p>Comparing the statements in both columns shows that figure 3(a) is self-dual and cannot be the dual of figure 3(b). We leave it to the reader to show that figure 3(b) is also self-dual. In both cases the cube and octahedron have a common centre, so reciprocation in a sphere can produce an octahedron and a cube simultaneously, and both compounds are actually self-reciprocal. It is interesting to note that if you do perform reciprocation in a sphere, then the resulting figure is rotated. In figure 3(a), it is rotated 60° about the common 3-fold axis. In figure 3(b), it is rotated 45° about the common 4-fold axis.</p> <hd id="AN0018528183-11">4.3. The Grünbaum and Shephard examples</hd> <p>Grünbaum and Shephard [<reflink idref="bib5" id="ref13">5</reflink>] consider several polyhedra and discuss the possibility of constructing their duals. Their first figure shows a pair of polyhedra which they state to be duals of each other. It is easy to see that they are combinatorial duals since these polyhedra are topologically equivalent to the dodecahedron and icosahedron. Figure 4(a) consists of twelve (non-convex) pentagons with three at each vertex and figure 4(b) of twenty triangles with five faces at each vertex. Although the article sets out to demonstrate that in general there is no correspondence between the convexity properties of a polyhedron and its dual, in this case they comment there are analogous symmetry and convexity properties, and leave the impression that the examples could be duals in some stronger sense, but they are certainly not reciprocal.</p> <p>Graph: Figure 4. Grünbaum and Shephard examples.</p> <p>Figure 4(a) can also be considered in the following way as described by Ounsted [<reflink idref="bib12" id="ref14">12</reflink>] in a paper entitled 'An unfamiliar dodecahedron'. The construction of a dodecahedron (which goes back to Euclid) consists of adding a set of 'roofs' to a cube. If these roofs are inverted, so that they are subtracted from the cube instead of adding to it, then the result is figure 4(a). For this reason, we will refer to this polyhedron as the inverted dodecahedron.</p> <p>Figure 4(b) can be considered as an icosahedron where three pairs of edges, defining orthogonal planes, have been pushed towards the centre. Jessen [<reflink idref="bib13" id="ref15">13</reflink>] wrote a paper about orthogonal icosahedra and so it is generally known as the Jessen icosahedron. We have drawn it slightly differently to the illustration in Grünbaum and Shephard [<reflink idref="bib5" id="ref16">5</reflink>], not only to make it more easily understood, but also deriving it from a regular icosahedron, so that when we create the reciprocal (figure 16),we get a polyhedron that is related to a regular dodecahedron. We have done no more than make a shallower depression than they have.</p> <p>Grünbaum and Shephard say that 'there is no difficulty finding topological complexes that are duals of any given polyhedron, but finding a <emph>dual polyhedron</emph> (their italics) is a much more illusive goal' [<reflink idref="bib5" id="ref17">5</reflink>]. They go on to show that in general this goal is unattainable if the dual is required to satisfy the usual definition of a polyhedron. Earlier on, however, they say [<reflink idref="bib5" id="ref18">5</reflink>] 'reciprocation is the <emph>only</emph> (their italics) known method of actually constructing a polyhedron P* dual to a given polyhedron P', and we find it curious that they did not actually try to develop this approach, because as we show later, it yields interesting and illuminating results. It shows that figures 4(a) and 4(b) are not reciprocals, although their reciprocals are interesting in their own right and are shown later (figures 15 and 19).</p> <hd id="AN0018528183-12">5. A geometrical construction for reciprocals</hd> <p>There is a geometrical construction for the duals of polyhedra attributed to Dorman Luke which is mentioned in Cundy and Rollett [<reflink idref="bib14" id="ref19">14</reflink>] and described in detail in Wenninger [<reflink idref="bib11" id="ref20">11</reflink>]. It is not an easy method to use and is only suitable for convex polyhedra being reciprocated in a sphere. It relies on taking the vertex figure (the polygon obtained by joining points along the edges around a vertex at a fixed distance from the vertex) and circumscribing the polygon. Tangents are then drawn at each vertex of the polygon (these are the polar lines of these points in the circle) to construct the reciprocal polygon. This method gives the shape of the faces of the reciprocal polyhedron, but it does not give the spatial coordinates and much more work is required to create three dimensional models in a CAD system. Because it is so restricted, and too cumbersome to allow us to explore reciprocals quickly and easily, we refer the reader to Wenninger [<reflink idref="bib11" id="ref21">11</reflink>].</p> <hd id="AN0018528183-13">6. Determining the reciprocal by calculation</hd> <p>Cartesian coordinates in Euclidean space use three variables <emph>x</emph>, <emph>y</emph>, and <emph>z</emph> to denote the position of an absolute position of a point in space relative to an origin. Projective geometry is not normally concerned with absolute positions, but in geometrical properties which are invariant under projection, so that, for example, a circle, an ellipse, a parabola and a hyperbola are all projectively equivalent.</p> <p>To work in projective space, we need to use homogeneous coordinates. Since we are using this coordinate system as a tool, we will only give some basic background for their use, and simplify the concepts for our needs without loss of rigour. We suggest that the reader goes to a book on projective geometry such as Coxeter [<reflink idref="bib10" id="ref22">10</reflink>] or Maxwell [<reflink idref="bib15" id="ref23">15</reflink>] for a detailed explanation.</p> <p>Homogeneous coordinates extend the Euclidean concept of a plane and space to infinite or ideal positions. If we are totally concerned with the finite, we can work with them so as to maintain an equivalence. Using homogeneous coordinates the position of a point in a plane is uniquely defined by the ratio of three coordinates and in space by four. So in a plane a point's position has three coordinates <emph>x</emph>, <emph>y</emph>, and <emph>z</emph>. To convert homogeneous coordinates of a two dimensional point (<emph>x</emph>, <emph>y</emph>, <emph>z</emph>) to Cartesian coordinates, simply divide all coordinates by the <emph>z</emph> value to give (<emph>x</emph>/<emph>z</emph>, <emph>y</emph>/<emph>z</emph>, 1).</p> <p>One of the beauties of using homogeneous coordinates is that the dual behaves in the same way. For example a line in the plane can be also be considered as the ratio of three coordinates (<emph>X</emph>, <emph>Y</emph>, <emph>Z</emph>). Then one equation surfices to describe dual situations. For example, the following 'linear' equation can take dual meanings:</p> <p>Graph</p> <p></p> <ulist> <item> 1. if <emph>X</emph>, <emph>Y</emph> and <emph>Z</emph> are constant, it describes a line as a range of points (all the points on the line)</item> <p></p> <item> 2. if <emph>x</emph>, <emph>y</emph> and <emph>z</emph> are constant, it describes a point as a pencil of lines (all the lines though the point)</item> </ulist> <p>For our purposes, dealing with points and planes in space, we need four coordinates for either. In common with the usual convention in writing homogeneous coordinates we will now use (<emph>x</emph><subs>1</subs>, <emph>x</emph><subs>2</subs>, <emph>x</emph><subs>3</subs>, <emph>x</emph><subs>4</subs>) to denote a point or a plane as we choose. This has benefits in being extensible to any dimension with the added benefits that you can quickly see which dimension you are working in. It also makes computer programs easier to write. Lines in polyhedra are sides of faces; they are drawn by computer as joins of points and polygons are specified as an ordered array of points.</p> <p>In programming terms, the duality of using (<emph>x</emph><subs>1</subs>, <emph>x</emph><subs>2</subs>, <emph>x</emph><subs>3</subs>, <emph>x</emph><subs>4</subs>) to denote a point or a plane as we choose means we can use one subroutine to perform operations on points and planes in space, (and similarly on points or lines in the plane) as will become clear in the algorithm for calculating reciprocals. Projective geometry using homogeneous coordinates is very common in computer graphics; for example see Penna and R Patterson [<reflink idref="bib16" id="ref24">16</reflink>].</p> <hd id="AN0018528183-14">6.1. Data used in the algorithm</hd> <p>To calculate the reciprocal, we need the following sets of information which are specified as a series of arrays.</p> <p></p> <p> <ephtml> <table><tbody valign="top"><tr><td><bold>Data</bold></td><td><bold>Used for</bold></td></tr><tr><td>coordinates of vertices</td><td>position of vertices</td></tr><tr><td>vertices in a face</td><td>plane coordinates of faces</td></tr><tr><td>order of faces around a vertex</td><td>order of vertices in a reciprocal face</td></tr></tbody></table> </ephtml> </p> <hd id="AN0018528183-15">6.2. Algorithm for computing the reciprocal</hd> <p>Since CAD programs can perform visualisation with the ability to alter the view, perform hidden line resolution and so on, we have programmed to write data which is in the CAD program format. The major part of the program for creating the file for loading into a CAD program is concerned with loading the data describing the original polyhedron and writing it out in the required format. This can take so many forms (and will change in the future), so we will only provide the essentials part of the program as the following steps in pseudo-code, together with the algebra for the calculations. Not everyone has access to a CAD system, but the VRML language is an alternative whose viewer is obtainable free on the internet (from the Cosmo software or Cortona sites) and the file is simple and easy to edit manually. The calculations themselves are simple enough to be made in a spreadsheet. So we have provided an appendix on how to work with a combination of spreadsheet and VRML which could easily be used in a classroom environment. The algorithm is as follows:</p> <p>Note that to determine the coordinates of a face, you only need three points, just as three planes uniquely determine a point. Care must be taken to choose three points that are not all in a line, otherwise the plane cannot be defined. If the face is degenerate such that there are only three points which lie in a line, then the dual of the face is also degenerate as three planes in a line. If the face is degenerate so that it shrinks to a single point, then the dual is degenerate as a single plane. In these cases, a vertex does not have a defined position, but is limited to a line or plane respectively.</p> <p>A critical aspect of this algorithm is finding the order of the corners of each face of the dual, which is then determined as a circuit, consistent with Grünbaum's definition of a polygon mentioned above. It is the same as the order of the faces around the corresponding vertex in the original polyhedron, which can be determined by observing that adjacent faces have a common edge.</p> <p>Because we are discussing duality, it would be equally valid to find the reciprocal of a point as a plane and assemble the polyhedron with more calculation. In practice, computer programs accept data as sets of points and so the above algorithm is more efficient and easier to program. It is also easier to visualise and write the algorithm because of this.</p> <hd id="AN0018528183-16">6.3. Projective calculations used in the algorithm</hd> <p>The three calculations required in the algorithm are relatively simple. Since they are tools we will only provide the results. Many standard books on projective geometry and homogeneous coordinates such as Smith [<reflink idref="bib17" id="ref25">17</reflink>] and Maxwell [<reflink idref="bib15" id="ref26">15</reflink>] can be accessed for further information.</p> <p>The simplest calculation is the transformation between Cartesian and homogeneous coordinates. We simply use the value 1 for the fourth coordinate. That is a point whose Cartesian coordinates have values (<emph>x</emph>, <emph>y</emph>, <emph>z</emph>) has homogeneous coordinates with values (<emph>x</emph>, <emph>y</emph>, <emph>z</emph>, 1).</p> <p>When we have to convert back, we normalise the fourth coordinate and then only use the first three as our Cartesian coordinates. That is:</p> <p>Graph</p> <p>Having found homogeneous coordinates for the vertices the next job is to find the coordinates of the planes containing the faces. If three points have coordinates (<emph>a</emph><subs>1</subs>, <emph>a</emph><subs>2</subs>, <emph>a</emph><subs>3</subs>, <emph>a</emph><subs>4</subs>), (<emph>b</emph><subs>1</subs>, <emph>b</emph><subs>2</subs>, <emph>b</emph><subs>3</subs>, <emph>b</emph><subs>4</subs>), (<emph>c</emph><subs>1</subs>, <emph>c</emph><subs>2</subs>, <emph>c</emph><subs>3</subs>, <emph>c</emph><subs>4</subs>) then the coordinates of the plane (<emph>x</emph><subs>1</subs>, <emph>x</emph><subs>2</subs>, <emph>x</emph><subs>3</subs>, <emph>x</emph><subs>4</subs>) they define are given by the following determinants:</p> <p>Graph</p> <p>Determining the reciprocal plane of a point with respect to a quadric is a standard procedure, and is covered in standard texts, for example Smith [<reflink idref="bib17" id="ref27">17</reflink>], so with conversion of the result to homogeneous coordinates, we get the result that the reciprocal plane of the point (<emph>a</emph><subs>1</subs>, <emph>a</emph><subs>2</subs>, <emph>a</emph><subs>3</subs>, <emph>a</emph><subs>4</subs>) with respect to the general conicoid:</p> <p>Graph</p> <p>(where the uppercase letters are constants) has coordinates</p> <p>Graph</p> <p>if we simplify this to a sphere centred at the origin</p> <p>Graph</p> <p>then the reciprocal plane has coordinates</p> <p>Graph</p> <p>where <emph>R</emph> is the radius of the sphere. Since we are using homogeneous coordinates the dual operation is identical in form, so if we are finding the reciprocal of a plane with coordinates (<emph>x</emph><subs>1</subs>, <emph>x</emph><subs>2</subs>, <emph>x</emph><subs>3</subs>, <emph>x</emph><subs>4</subs>), this gives us a pole with coordinates (<emph>x</emph><subs>1</subs>, <emph>x</emph><subs>2</subs>, <emph>x</emph><subs>3</subs>,–<emph>R</emph><sups>2</sups><emph>x</emph><subs>4</subs>) or Cartesian coordinates (<emph>x</emph><subs>1</subs>/–<emph>R</emph><sups>2</sups><emph>x</emph><subs>4</subs>, <emph>x</emph><subs>2</subs>/–<emph>R</emph><sups>2</sups><emph>x</emph><subs>4</subs>, <emph>x</emph><subs>3</subs>/–<emph>R</emph><sups>2</sups><emph>x</emph><subs>4</subs>).</p> <p>This result shows that the size of the reciprocating sphere has no effect on the shape of the reciprocal polyhedron but only on its size by virtue of –<emph>R</emph><sups>2</sups> being the factor which decides the size.</p> <hd id="AN0018528183-17">7. Results and some new polyhedra</hd> <p>Now we have a simple program to calculate the reciprocal of a polyhedron, we can explore quickly and easily by small variations in a way that would be difficult and tedious by conventional geometric methods. But first, to enable interpretation of the results, we need to consider some more general polyhedra.</p> <hd id="AN0018528183-18">7.1. Non-convex polygons and polyhedra</hd> <p>In his consideration of definitions of polyhedra, Cromwell [<reflink idref="bib2" id="ref28">2</reflink>] shows two cases of polyhedra which were recorded in 1830 by the mineralogist Hessel as counter-examples to Euler's formula which we have redrawn in figure 5.</p> <p>Graph: Figure 5. Hessel's counter examples to Euler's formula.</p> <p>Most definitions of a polyhedron, such as Coxeter's, avoid the difficulty by excluding them. However, if you were to make them as models (for example using bamboo skewers) they can be seen as being of different types. The one on the left would not be rigid, but could flex about edge BD. We need not consider polyhedra of this type. Polyhedra that have intersecting faces (such as figure 19) look rather like this, but the intersection is not an edge, and no flexing is possible. The one on the right (figure 5(b)) could be made so that AB, CD and EF were straight lines such that AFEB, CEFD and ADBC are intersecting quadrilaterals. The figure is constructed by joining polygons along their sides, and so qualifies as a polyhedron under the wider definition mentioned earlier. Point G is the point of intersection, but not a vertex. The polyhedron is thus a type of prism. It has three four-sided faces and two triangular ones, six vertices and nine edges and thus conforms to Euler's formula.</p> <p>The way the sides of the quadrilaterals intersect is analogous to the way the sides of a star polygon do. Polyhedra with intersecting faces using pentagrams and triangles are well known in the Kepler–Poinsot polyhedra. Such intersecting polyhedra will appear when we try to form reciprocal polyhedra.</p> <hd id="AN0018528183-19">7.2. More Grünbaum and Shephard 'duals'</hd> <p>This problem of intersection arises in Grünbaum and Shephard's next set of examples [<reflink idref="bib5" id="ref29">5</reflink>]. In their figure 13-2 (on p. 208) they show five polyhedra, which are apparently simple cases, and discuss their duality. We will start by considering the first pair of these in figure 6.</p> <p>Graph: Figure 6. A polyhedron and its reciprocal, taken from Grünbaum and Shephard.</p> <p>Figure 6(a) (their fig. 13-2a) consists of a polyhedron made from a cube with a square pyramid (part of an octahedron) erected on one face. We can visualize the dual by recalling that the dual of a cube is an octahedron and hence vice versa. The apex (point) of the pyramid gives rise to a new face (plane) of the dual (figure 6(b)), which is a square, since the vertex is 4-valent, and the edges have square symmetry. Four faces of the dual become trapezia instead of triangles, corresponding to the vertices where the pyramid joins the cube, which are 4-valent. The four points at the base of the cube become four triangles meeting at the point which is dual to the base of the cube. The resulting dual shown in figure 6(b) (which is Grünbaum and Shephard's figure 13-2b), they point out, gives a pair which is combinatorially/projectively equivalent, and so they are self-dual. In figure 6(a) we have added a point to the cube while at the same time losing a face. This means in the dual (figure 6(b)) we have added a plane to an octahedron which has lost a vertex.</p> <p>Note that the reciprocal has been calculated with respect to a sphere centred in the middle of the cube.</p> <p>If the apex is moved along the axis of symmetry, its dual (the square plane at the top in figure 6(b)) will move also, intersecting the octahedron at different places to give a varying square face. The higher the apex, the nearer the face to the mid-plane of the octahedron. If the apex is lowered so that it sits in the top face of the cube, then the plane cuts the octahedron at its vertex, and the four corners of the square, which has sides of zero length, are all at the same point.</p> <p>If the apex is moved further towards the centre of the cube (giving figure 7(a) which is like Grünbaum and Shephard's figure 13-2c), the plane moves further away from the mid-plane of the octahedron, and a polyhedron is produced having faces that are self-intersecting trapezia.</p> <p>Graph: Figure 7. Another Grünbaum and Shephard example, with its reciprocal.</p> <p>The reciprocal in figure 7(b) needs to be thought of in the same way as the Hessel polyhedron in figure 5(b). The 'extra vertex' which looks as if it is the top vertex of an octahedron is something we can perceive although it is not a vertex of the polyhedron we have constructed. Its dual is the top 'extra face' of the cube in figure 7(a) which we can imagine but not touch.</p> <p>If the apex is moved to the centre of the cube, and we continue to create the reciprocal with respect to a sphere centred at the centre of the cube, then the dual of the apex moves to infinity. Figure 8(a) shows one representation of this case. It would be equally valid to draw the infinite lines to the square at infinity above the half octahedron.</p> <p>Graph: Figure 8. The reciprocal as the apex in Figure 7(a) is depressed further.</p> <p>If we take the apex down past the centre of the 'cubic' part, then the square in the dual comes back from underneath (figure 8(b)). The self-duality then becomes more evident. If the apex is taken below the base of the cube then the dual is similarly self-dual. Figures 8(c) and 8(d) show the same polyhedron without, and then with, hidden line removal. Figure 8(c) has had no line thickness adjustments to attempt to show the way the lines are hidden, neither does it show the lines resulting from intersection with the square visible in figure 8(d).</p> <p>These examples illustrate an aspect of the reciprocation process applied to polyhedra that is usually ignored. Reciprocation is a mapping between points, and infinite lines and planes, but polyhedra consist of vertices, and (usually) finite line-segments and faces. Normally the natural identification of an edge with the finite part of the line between two vertices goes unremarked, and it occurs automatically when the reciprocation algorithm is executed, but cases such as figure 8(a) make it obvious that a choice is being made. It is also clear that there is a discontinuity at this stage, when the set of four semi-infinite edges flips from pointing upwards to pointing downwards, and the edges that are the finite parts of these lines in figure 8(b) correspond to the infinite parts that are ignored in figures 6(b) and 7(b).</p> <p>In figure 7(b), we have polygons which are self-intersecting. This is purely a result of where the quadric (in this case a sphere) is positioned when reciprocating. By moving the polyhedron in figure 7(a) up until the centre of the sphere lies outside the polyhedron, the self-duality is more evident (figure 9). There are now no self-intersecting polygons.</p> <p>Graph: Figure 9. Changing the centre of reciprocation in Figure 8(d).</p> <p>The polyhedron in figure 7(a) does not have the complete symmetry of the cube, only a fourfold symmetry about the vertical axis. We could make a polyhedron in which there are dimples on each face of the cube. If they all reach the centre of the cube then the squares in the reciprocal go to infinity if the centre of our reciprocating sphere is at the centre of the cube. However, if we only make the dimples slight, so that we get a symmetrical version of figure 7(b), and create the reciprocal, the polyhedron does exist in the literature. It is figure 104 in Brückner [<reflink idref="bib18" id="ref30">18</reflink>] shown in our figure 10(a).</p> <p>Graph: Figure 10. Brückner polyhedra.</p> <p>We can also consider what happens to reciprocals of polyhedra formed by adding square pyramids to each face of a cube when the apices of the pyramids are moved symmetrically. The measurements for the height of the apices are given relative to a starting cube of side 2 units.</p> <p>The reciprocal begins as an octahedron with all six vertices truncated, forming square faces (figure 11(b)) where the apices in figure 11(a) are 1.15 units above the faces. As the apices of the square pyramids are raised, the squares of the truncated octahedron move down, at one stage forming a regular truncated octahedron (figure 11(d)) when the apices move to 1.5 units as in figure 11(c) which corresponds to Catalan's tetrakis hexahedron. Continuing to raise the apices to 2 units as in figure 11(e),the triangular faces of the pyramids become coplanar in pairs, forming a rhombic dodecahedron with each face divided into two triangles. In the reciprocal the vertices of the squares touch to form a polyhedron that looks like a cuboctahedron (figure 11(f)), but the triangular faces are actually hexagons, having three of the sides of zero length, and what appear as single vertices in the figure are actually pairs of vertices. Raising the apices further is like 'folding' the rhombs of the rhombic dodecahedron' about the cube edges so that the edge is no longer convex. In the two examples shown, figure 11(g) has the apex at 2.5 units and figure 11(h) at 3.</p> <p>Graph: Figure 11. Cubes with square pyramids added to their forces with their reciprocals.</p> <p>Note that polyhedra like these also exist in Brückner [<reflink idref="bib18" id="ref31">18</reflink>]. The one in figure 10(b) is figure 1 in table XI of Brückner.</p> <hd id="AN0018528183-20">8. Convexity and figure 4(a) and 4(b)</hd> <p>In the earlier discussion we noted Grünbaum and Shephard's consideration of the use of convexity as a means of obtaining duals such as figure 4(a) and 4(b), and their conclusion that 'convexity properties cannot be preserved in duality', which they illustrate by considering the polyhedron in (our) figure 7(a). They say that it 'has four non-convex edges meeting at a vertex, so a dual <emph>ought</emph> (their italics) to have four non-convex edges bounding a quadrangular face', and observe that no such polyhedron exists. Indeed there are no known rules to predict the convexity properties of parts of the dual from parts of the original polyhedron, and even the general observation that convex polyhedra have convex duals breaks down if there are no restrictions on the centre of the reciprocating sphere. For example, figures 9 and 7(b) are both duals of figure 7(a) because they have been created as reciprocals, and they differ because the position of the reciprocating sphere is different. Similarly, the set of 'octahedra' in figure 12 shows the effect of reciprocating a cube by successively moving the centre of the reciprocating sphere up from the centre of the cube.</p> <p>Graph: Figure 12. Reciprocal of a cube with different positions of reciprocating sphere.</p> <p>As the centre of the reciprocating sphere moves up, the top half of the octahedron becomes more elongated. The top vertex then jumps over infinity and as in figure 8 we use the finite line segments. The faces that originally formed the convex base now become a concave top. Thus different duals of the same polyhedron may have different convexity properties.</p> <p>We are now in a position to look at the inverse dodecahedron (figure 4(a)) for which Grünbaum and Shephard gave the misleading dual (figure 4(b)) and to determine the reciprocal of figure 4(b). In the following, the duals are created as reciprocals with respect to a sphere whose centre is at the centre of symmetry of the two polyhedra.</p> <p>Some properties of the dual of the inverse dodecahedron (figure 4(a)) can be deduced by comparing it with some well-known related polyhedra that have accepted duals. It can be considered as a great stellated dodecahedron (figure 13(a)) with twelve of the spikes removed. Twelve of the crossing points of the star pentagons, which are usually not considered to be vertices and so have no corresponding faces in the dual, the great icosahedron (figure 13(b)), become true vertices. The dual will have eight planes of faces in common with the great icosahedron, corresponding to the eight spikes that are not removed, and since these vertices form the vertices of the cube that is the convex hull of the inverted dodecahedron, the dual faces surround an octahedral core.</p> <p>Graph: Figure 13. Great stellated dodecahedron and great icosahedron.</p> <p>Viewing the great stellated dodecahedron as an icosahedron with triangular pyramids (the spikes) stuck on each face makes it clear that the other twelve vertices of the inverted dodecahedron are common with an icosahedron, and so they are dual to faces lying in the planes of a dodecahedron. Since the inverted dodecahedron has only one type of face, its dual has one type of vertex, which we have already seen to be those of the great icosahedron, and the vertices of the great icosahedron are those of its convex hull, an icosahedron. So these twelve faces in the dual are those of a polyhedron that is a facetted icosahedron and a stellated dodecahedron. They are in fact part of a great dodecahedron (figure 14).</p> <p>Graph: Figure 14. Great dodecahedron.</p> <p>So, we can say that the complete dual has twelve (icosahedral) vertices and twenty faces, eight are equilateral triangles from the great icosahedron, twelve are acute isosceles triangles derived from the pentagonal faces of a great dodecahedron. Although it is still difficult to visualise it, the reciprocation algorithm will generate pictures of the dual, and the foregoing discussion will help in interpreting them.</p> <p>At first glance the result is not straightforward, and there appear to be faces and edges missing. Figure 15 is a stereoscopic pair to aid examination. There is a similar 'wedge' shape at the back, completing the symmetry, which is hidden.</p> <p>Graph: Figure 15. Reciprocal of inverted dodecahedron (figure 5 left).</p> <p>There are points that are not vertices, for example where three faces intersect in the dimples either side of the 'wedge' shapes. The true twelve vertices are at the tips of the wedges.</p> <p>There appear to be only four faces meeting at a vertex, and because of the intersections only parts of some faces can be seen. Counting the visible parts gives a total of 36, so some parts must belong to the same face.</p> <p>When a dual is calculated by reciprocation, there is an added bonus of numerical data, for example information about the two types of triangle. The set of twelve are isosceles triangles with two angles being 72° and sides in the ratio of the golden section. The shortest sides of these triangles are the sharp edges of the wedges. The other triangles are equilateral triangles with three parts of each of them visible. The edges of the twelve isosceles triangles sit in the faces of other similar triangles. They can be drawn in as in figure 16.</p> <p>Graph: Figure 16. Reciprocal of inverted dodecahedron with complete edges.</p> <p>This now shows the five faces in each vertex. It also shows that the edge going to the vertex in the dimple formed by intersections is also a line formed by the intersection of two equilateral triangles. If this were not so, too many edges would meet at a vertex. Also, since edges of the polyhedron are common to two faces, the equilateral triangles must also lie in the faces of the isosceles triangles. The edges of the equilateral triangles are the same as the long edges of the isosceles triangles. This all suggests a complex internal structure to this dual with the 'virtual' octahedron buried in the centre.</p> <p>Figure 17 shows the result of removing all the isosceles triangles to leave the eight equilateral triangles as a stellation of the octahedron. This does not fall within the usual definition of stellations, since the sides of the triangles are not common edges; the only closed stellation of the octahedron is the stella octangula.</p> <p>Graph: Figure 17. Reciprocal of inverted dodecahedron, internal structure.</p> <p>The octahedron can only be seen by slicing to get to the interior as shown in figure 18.</p> <p>Graph: Figure 18. Reciprocal of inverted dodecahedron, section to show octahedron.</p> <p>Thus the dualization considered by Grünbaum and Shephard was wide of the mark, and it is not even necessary to construct the reciprocal to deduce that the polyhedron is inconsistent with the well-known duality relations between the Kepler–Poinsot polyhedra.</p> <p>We have only touched on the properties of the inverse dodecahedron and its dual (which we believe to be a new polyhedron) here and they are obviously polyhedra which deserve more study.</p> <p>The dual of the Jessen icosahedron in figure 4(b) is much simpler. Earlier, we said that the drawing is different from the one shown by Grünbaum and Shephard. We have chosen to construct it from two sets of triangles as follows. The external triangles are equilateral. Those where there is a concavity are isosceles triangles with two angles of 36°, that is a triangle with sides in the golden section. Figure 4(b) is a regular icosahedron with wedges cut out of it perpendicular to parallel edges, and so we might expect to see a regular dodecahedron with some additions at edges. This is indeed the case as figure 19(a) shows.</p> <p>Graph: Figure 19. Reciprocal of Jessen icosahedron (figure 5 right).</p> <p>Figure 19(b) shows how the pentagons of the dodecahedron have been extended and intersect to form an equilateral (though not equiangular) pentagon. We also believe this to be a new polyhedron.</p> <hd id="AN0018528183-21">9. Conclusions</hd> <p>The term 'duality', meaning a mapping between points and planes (or vertices and faces), generally occurs in geometries where these are fundamental objects. In projective geometry they occur symmetrically in the axioms, so that duality is a natural consequence. Some of the problems that arise when considering dual polyhedra occur when there is some further structure. For example Grünbaum and Shephard [<reflink idref="bib5" id="ref32">5</reflink>] define polyhedra as manifolds and go on to show that in some circumstances the dual of a polyhedron does not fit this definition. Another example is provided by figure 2. Different faces being coplanar seems unremarkable, since they occupy different positions in space. The dual situation, different vertices being coincident, is certainly contrary to common usage, and is very counter-intuitive.</p> <p>Further problems occur because the concept of 'duality' carries a different meaning in different contexts. It is important that there should be no ambiguity in its application if misunderstandings are to be avoided, and care is needed to be sure of obtaining logically consistent results.</p> <p>With modern computer graphics and CAD packages it is easy to implement an algorithm for reciprocation, which not only provides a useful way to check ideas about polyhedra and their duals, but also offers much potential for further study and the discovery of new ones. We hope to provide further examples soon.</p> <hd id="AN0018528183-22">Appendix. Using VRML to view polyhedra</hd> <p>Not everyone can program or use a CAD program, but in learning mathematics it is reasonable to assume that a spreadsheet could be used to perform calculations and a simple VRML file can be used to view the result with free VRML plugins for Internet browsers being available on the internet for different platforms. VRML (Virtual Reality Markup Language) has an easy to use file format which can be written in any text editor. Providing simple templates into which calculated data can be edited overcomes any need for learning to program the language. The result can be viewed in minutes and the polyhedron manipulated in real time. The following example shows how to analyse a simple polyhedron and use it in a VRML file which can serve as a template.</p> <p>As well as being a good training in analysis of data, the calculation in a spreadsheet provides a useful exercise of organisation of the spreadsheet (arranging data to make it easy to copy formulae) and the use of features like determinants. This spreadsheet exercise is left to the reader, since the algorithm is very easy to implement.</p> <p>The example uses a square pyramid with base as shown in figure 20 with Cartesian coordinates:</p> <p></p> <p> <ephtml> <table><tbody valign="top"><tr><td>A (1, 1, 0)</td></tr><tr><td>B (1, −1, 0)</td></tr><tr><td>C (−1, −1, 0)</td></tr><tr><td>D (−1, 1, 0)</td></tr><tr><td>E (0, 0, 1)</td></tr></tbody></table> </ephtml> </p> <p>Graph: Figure 20. Square pyramide used in VRML example.</p> <p>In order to keep the calculation simple, the reciprocating sphere is left at the origin and an offset added to move the polyhedron. With the coordinates above there is no offset, so the plane ABCD goes though the centre of the sphere and its pole goes to infinity. The following table shows the calculated coordinates of the poles of each plane.</p> <p></p> <p> <ephtml> <table><tbody valign="top"><tr><td><italic>offset (0,0,1)</italic></td><td><italic>offset (0,−2,−2)</italic></td></tr><tr><td>(0, 0, −1) (−0.5, 0, −0.5) (0, −0.5, −0.5) (0.5, 0, −0.5) (0, 0.5, −0.5)</td><td>(0, 0, 0.5) (1, 0, 1) (0, 0.3333, 0.3333) (−1, 0, 1) (0, 1, −1)</td></tr></tbody></table> </ephtml> </p> <p>With an offset in the <emph>z</emph> direction, as in the first set of coordinates, the self duality is evident as a similar pyramid is produced. The second set of coordinates, with an offset not along the <emph>z</emph> axis the base is no longer square. The result is shown in figure 21.</p> <p>Graph: Figure 21. Screen dump from VRML viewer.</p> <p>The template VRML file is as follows:</p> <p>It is not necessary to understand the template in detail in order to use it. The important part for the purposes of this paper is to see that the coordinates are entered as a table followed by a table which indexes each plane. So the quadrilateral is the ordered set of points 1, 2, 3, 4 which is then terminated with a −1, as are each of the triangular planes.</p> <p>There is not space to describe more sophisticated use of VRML with Java scripts to allow dynamic movement of the polyhedron relative to the sphere, either animated or under user control.</p> <ref id="AN0018528183-23"> <title> References </title> <blist> <bibl id="bib1" idref="ref1" type="bt">1</bibl> <bibtext> HoldenA1971Shapes, Space and SymmetryNew York & LondonColumbia University Pressrepublished by Dover</bibtext> </blist> <blist> <bibl id="bib2" idref="ref2" type="bt">2</bibl> <bibtext> CromwellP1997PolyhedraCambridgeCambridge University Press</bibtext> </blist> <blist> <bibl id="bib3" idref="ref3" type="bt">3</bibl> <bibtext> de Villiers, M. 1998. Dual generalisations of Van Aubel's theorem. Mathematical Gazette, 495: 405–412.</bibtext> </blist> <blist> <bibl id="bib4" idref="ref4" type="bt">4</bibl> <bibtext> Bakos, T. 1959. Octahedra inscribed in a cube. Mathematical Gazette, 43: 17–20.</bibtext> </blist> <blist> <bibl id="bib5" idref="ref5" type="bt">5</bibl> <bibtext> GrünbaumBShephardGC1988Duality of PolyhedraIn: M. Senechal and G. Fleck (Eds)Shaping SpaceBostonMABirkhäuser</bibtext> </blist> <blist> <bibl id="bib6" idref="ref6" type="bt">6</bibl> <bibtext> CoxeterHSM1973Regular Polytopes, Dover</bibtext> </blist> <blist> <bibl id="bib7" idref="ref7" type="bt">7</bibl> <bibtext> GrünbaumB1993Polyhedra with hollow facesIn: T. Bisztriczky, P. McMullen, R. Schneider and A. Ivic'Weiss (Eds)Proc. NATO-ASI Conference on Polytopes: Abstract, Convex and ComputationalToronto, 1993DordrechtKluwer Academicpp. 43–70</bibtext> </blist> <blist> <bibl id="bib8" idref="ref8" type="bt">8</bibl> <bibtext> MöbiusAF1831Analytisch-geometrische Entwickelungen21831</bibtext> </blist> <blist> <bibl id="bib9" idref="ref9" type="bt">9</bibl> <bibtext> PedoeD1975Notes on the History of Geometrical Ideas II: the principle of DualityMathematics MagazineNov–Dec274277</bibtext> </blist> <blist> <bibtext> CoxeterHSM1955The Real Projective PlaneCambridgeCambridge University Presspp. 74–75</bibtext> </blist> <blist> <bibtext> WenningerMJ1983Dual ModelsCambridgeCambridge University Press</bibtext> </blist> <blist> <bibtext> Ounsted, J. 1978. An unfamiliar dodecahedron. Mathematics Teaching, 83: 46–47.</bibtext> </blist> <blist> <bibtext> Jessen, B. 1967. Orthogonal icosahedra. Nordisk Matematisk Tidskrift, 15: 90–96.</bibtext> </blist> <blist> <bibtext> CundyHMRollettAP1961Mathematical ModelsOxfordOxford University Press</bibtext> </blist> <blist> <bibtext> MaxwellEA1963The Methods of Plane Projective Geometry based on the use of General Homogeneous EquationsCambridgeCambridge University Press</bibtext> </blist> <blist> <bibtext> PennaMPattersonR1986Projective Geometry and its Applications to Computer GraphicsEnglewood CliffsNJPrentice Hall</bibtext> </blist> <blist> <bibtext> SmithC1901An Elementary Treatise on Solid GeometryLondonMacmillan</bibtext> </blist> <blist> <bibtext> BrücknerM1900Vielecke und VielflacheLeipzigTuebner</bibtext> </blist> </ref> <aug> <p>By P. Gailiunas and J. Sharp *</p> <p>Reported by Author; Author</p> </aug> <nolink nlid="nl1" bibid="bib10" firstref="ref10"></nolink> <nolink nlid="nl2" bibid="bib11" firstref="ref11"></nolink> <nolink nlid="nl3" bibid="bib12" firstref="ref14"></nolink> <nolink nlid="nl4" bibid="bib13" firstref="ref15"></nolink> <nolink nlid="nl5" bibid="bib14" firstref="ref19"></nolink> <nolink nlid="nl6" bibid="bib15" firstref="ref23"></nolink> <nolink nlid="nl7" bibid="bib16" firstref="ref24"></nolink> <nolink nlid="nl8" bibid="bib17" firstref="ref25"></nolink> <nolink nlid="nl9" bibid="bib18" firstref="ref30"></nolink>
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  Data: Duality of Polyhedra
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  Data: <searchLink fieldCode="SO" term="%22International+Journal+of+Mathematical+Education+in+Science+%26+Technology%22"><i>International Journal of Mathematical Education in Science & Technology</i></searchLink>. Sep 2005 36(6):617-642.
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  Data: Journal Articles<br />Numerical/Quantitative Data<br />Reports - Descriptive
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  Data: <searchLink fieldCode="DE" term="%22Logical+Thinking%22">Logical Thinking</searchLink><br /><searchLink fieldCode="DE" term="%22Computer+Graphics%22">Computer Graphics</searchLink><br /><searchLink fieldCode="DE" term="%22Computer+Simulation%22">Computer Simulation</searchLink><br /><searchLink fieldCode="DE" term="%22Thinking+Skills%22">Thinking Skills</searchLink><br /><searchLink fieldCode="DE" term="%22Geometric+Concepts%22">Geometric Concepts</searchLink><br /><searchLink fieldCode="DE" term="%22Geometry%22">Geometry</searchLink><br /><searchLink fieldCode="DE" term="%22Mathematics+Instruction%22">Mathematics Instruction</searchLink><br /><searchLink fieldCode="DE" term="%22Computation%22">Computation</searchLink><br /><searchLink fieldCode="DE" term="%22Numbers%22">Numbers</searchLink>
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  Data: Everyone is familiar with the concept that the cube and octahedron, dodecahedron and icosahedron are dual pairs, with the tetrahedron being self-dual. On the face of it, the concept seems straightforward; however, in all but the most symmetrical cases it is far from clear. By using the computer and three-dimensional graphics programs, it is possible to clarify the concept and explore new ideas. Moreover, it is an ideal topic for teaching clear logical thinking. (Contains 21 figures.)
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      – TitleFull: Duality of Polyhedra
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