Department of Mathematics, Technion - Israel Institute of Technology, Haifa, ISRAEL
Software Engineering Department, Ort Braude College, Karmiel, ISRAEL
Since a second purpose of any such course is to help develop geometric intuition and spatial imaginative powers, Euler's Theorem introduces one effortlessly in the realms of geometric creativity, by its natural generalizations into two directions: (a) high genus and non-orientable surfaces such as the Klein Bottle and The Projective Plane (thus representing an excursum in non-Euclidian geometries and also into patterns and tilings); and (b) Star and Uniform Polyhedra. (We shall indicate how and where these objects and ideas arise.)
Moreover, this approach allows for a perfect integration between the metric and topological aspects, thus presenting the dialectics of Mathematics, represented by the computational-arithmetic/figurative-geometric dichotomy.
We shall show in what manner the path outlined herein facilitates a potent yet flexible Curriculum: one can put more accent on Graph Theory or, alternatively, on "rigid" geometry, on incidence and topology or rather upon trigonometric computations. Through Graphs one attains such applications as: organization graph of architectural structures (introducing duality and planarity), traversability, routing, connectedness, classification of architectural plans. Even the very approach to the proofs shines light on different aspects of Geometry.
Upon these manifold proofs and the various, kaleidoscopic facets of Geometry they illuminate, we shall dwell (alas, briefly!) in the following section.
2 THE MATHEMATICS
Our proof of choice, the first one to present our students, is basically Euler's original one, based upon first triangulating the polyhedron, and then one by one removing the resulting triangles, while showing that the number V -E +F remains invariant during this process. As a special didactic trick, we present Gamow's variant of this proof, (see [Gamow 1988] ). In Gamow's presentation the polyhedron's edges are viewed as dams, the polyhedron as an isle, and the exterior of the resulting map as the sea. As a special flourish, we present this as a last defence against a Spanish Armada invading the Netherlands, sometime in a mythical sixteenth century (see Fig. 1). The advantage of this proof resides in it simplicity and in the fact that it introduces the extremely important notion of triangulation. But it provides us with even more depth: since an essential step in the proof is the removal of the upper face f0 (see Fig. 1) and the projection of the reminder of the polyhedron on the plane , this allows us to discuss the Stereographic Projection.
Fig. 1. The First Proof
(Considering projections can also lead to Steiner's proof of Euler's Theorem ; see [Sommerville 1942, 142] ). Moreover, faces as homeomorphic images of the disk and the topological concept of map (as opposed to that of a mere graph embedding) come under study naturally. From this point on, the possibilities are practically unlimited. The most direct road leads to counting the regular tilings of the plane and to that of Archimedian solids and semiregular tilings of the plane.
But the main advantage lies, first and foremost, in the fact that one can introduce the notion of graph in a light, natural manner, and with them a variety of problems of great significance in the theoretical setting and of vast practical importance, such as planarity and duality (via "the five brothers problem" [Baglivo and Graver 1988] and the "gas-light-water graph" [Baglivo and Graver 1988; Tietze, 1965] and thus to the Theorems of Kuratowsky and Whitney [Baglivo and Graver 1988; Tietze, 1965]. Trees are the simplest truly interesting graphs and they provide us with a third Proof of Euler's Theorem: Von Staudt's proof  based upon spanning trees. Paradoxically, the benefit of this proof is that it does not extend to surfaces of positive genus -- thus allowing yet a different insight into the topology of surfaces. Applications of Euler's Theorem, such as Kuratowski's Theorem are now natural -- through the inequality E<3V-6 ("The little inequality that could"). Its dual inequality E<3F- 6 provides us with an easy proof of the Six Color Theorem (and with an excuse to wander into a discussion of the Four Color Theorem).
Planarity and maps conduct us immediately to consider other surfaces than the plane or the sphere, such as the torus and compact, orientable surfaces of higher genus, and also non-orientable surfaces, in particular the Möbius strip, the Klein bottle and the projective plane (see Fig. 2).
Fig. 2. Topological Models
Also, as an immediate -- yet somewhat collateral -- development, maximal planar graphs and Fundamental Architectural Arrangements [Baglivo and Graver 1983, 115-119] ensue. Yet another Proof is needed if one considers further generalizations, such as Star Polyhedra and the even more general Uniform Polyhedra  (see Fig. 3).
Fig. 3. Star and Uniform Polyhedra
For these one has to appeal to the Spherical Area proof (or Legendre's Proof) [Coxeter 1963].
Fig. 4. An Archimedian Solid and its Spherical Counterpart
While a formal treatment of surface coverings is, of course, out of the courses scope, its rudiments are highly intuitive. Moreover, a much more technical instrument becomes now tangible: a numerical solution to compute the elements (sides, dihedral angles, radii, etc.) of a Uniform Polyhedron. The method above - based upon a many variable adaptation of Newton's Tangent Method for solving equations - is the one developed in [Har'el 1993]. Even if in the beginning the students tend to oppose (at an instinctive, self-preserving level) the meanders of its computation, in the end, the concreteness and clarity of the numerical results obtained rewards them with a better insight and understanding of the objects they studied until this point only on a descriptive, visual way. This is an intrinsic result sprouting from the very depths of mathematical thought and understanding (but we shall resume this discussion again in the last section).
3 DIDACTICAL ASPECTS
Moreover, the focus on Polyhedra facilitates the use of Constructive Projects as Marking Tool, thus addressing the demiurgical skills of the students, and adding yet another unifying, summarizing tool at the Course end. Indeed, Architecture students have not only the ability and the habit to express themselves in a constructive manner, they do posses propensity, the proclivity for this Renaissance type of expression: material and spiritual, combined (see Fig. 4).
Fig. 5. A Cornucopia of Models
Also, it is this author's firm belief that "People are much smarter when they can use their full intellect" ([Thuston, 1990] ). Even more: creative freedom and trust are stronger moving forces and better guaranties of novel, original ideas (for the final projects) then some constrictive exam frame. Trigonometric equations may be tedious and boring, but the become your equations when they help finalize a project. (See, for example the two Origami polyhedra of Fig. 6).
Fig. 6. Two Origami Polyhedra
4 BEYOND EULER'S THEOREM
It is stated in [Consiglieri and Consiglieri 2003] that "...mathematics does not lead to emotional forms but abstract ones; that responsibility belongs to aesthetics". This statement may conform to the common feeling of practicing artists. Nevertheless, it contradicts the very cultural tradition that resides at the inner core of the choice of any Course in Mathematics for Architects, the fact that "Classical mathematics is a quest for structural harmony" [Gromov 1998]. As yet another of the Titans of Contemporary Mathematics confesses: "Mathematics has a remarkable beauty, power and coherence, more than we could have ever expected" [Thurston 1990] and "Mathematics is like a flight of fancy " [Thurston 1990]. And again, in Freudenthal's words: "... mathematics is an interplay of content and form". Yet couldn't this serve as a concise, functional definition of Art?
And even if we restrict ourselves to a more mundane, practical level: if a course offers its listeners a plethora of examples, a gallery of fantastic forms, to fertilize, to help germinate and to serve as nutrient for growth of Pure Art, wouldn't it served his purpose? With this question that contains a hope, a belief in a positive answer, we conclude our essay.
 In an even earlier incarnation it also comprised elements of Differential Geometry of Surfaces -- see [Har'el 1985]. return to text
 This notion requires some mathematical formalism: Let X be a topological space. The k-th Betti number ßk of X is defined as the rank of the k-th homology group of X: ßk = rankHk . return to text
 Let S be a compact surface. Then , where K represents the Gauss curvature and c(S) is the Euler characteristic of S. return to text
 See [Arnold 1997] and Section 3 below. return to text
 Of nested cosmological regular polyhedra. return to text
 The geometric, angle based proof, is also presented, but more like an afterthought, an instructive variation. return to text
 A proof that made this author to chose Mathematics as his profession -- this, and Cantor's Diagonal Proof. return to text
 Thus reducing the problem to proving that, for the new map V -E +F =1. return to text
 The basic idea of Steiner (and Lhuilier's) proof is to project the polyhedron orthogonally on a plane, obtaining a polygon covered twice by a set of polygons, then express the sum of the angles of these polygons in terms of V, E, F. return to text
 The main steps of Von Staudt's proof are as follows: Build the spanning tree of the vertex set - the number of its edges will be E1 = V-1. Construct also the dual tree (of the face set) - the number of edges will be E2 = F - 1. Since the trees are disjoint we have: E = E1 + E2 = V + F - 2, i.e., V-E+F =2. return to text
 And, of course, Regular and Uniform Polyhedra direct us to the study of Symmetry Groups. return to text
 Indeed, mathematicians refer to their object of study (or should we say "passion"?!) in aesthetic terms: "What a Beautiful Theorem!", a "lovely idea", a "nice", "beautiful" or even "elegant" proof. return to text
Baglivo, Jenny A. and Graver, Jack E. 1983. Incidence and symmetry in design and architecture. Cambridge: Cambridge University Press.
Barr, Stephen. 1964. Experiments in topology. New York : Crowell.
Consiglieri, Luisa and Consiglieri, Victor. 2003. A Proposed Two-Semester Program for Mathematics in the Architecture Curriculum. Nexus Network Journal 5, 1: 127-134.
Coxeter, Harold S. M. 1961. Introduction to Geometry. New York: Wiley.
Coxeter, Harold S. M. 1963. Regular Polytopes. New York: Macmillan.
Crowe, Donald W. 1969. Euler's formula for Polyhedra and related topics. Pp. 178 in Excursions into Mathematics. New York: Worth Publications.
Cundy, H. Martyn and Arthur P. Rollet. 1961. Mathematical models. Oxford : Clarendon Press.
Francis, George K. 1987. A topological picturebook. New York : Springer.
Freudenthal, Hans. 1991. Revisiting mathematics education: China lectures. Dordrecht: Kluwer.
Gamov, George. 1988. 1, 2, 3 - infinity: facts and speculations of science. New York : Dover.
Gromov, Mikhael. 1998. Possible Trends in Mathematics on the Coming Decades. Notices of the A.M.S.: 45, 7: 846-847.
Har'el, Zvi. 1985. Geometry for Architects. Haifa: Technion. (in Hebrew)
______. 1993. Uniform Solution for Uniform Polyhedra. Geometria Dedicata, 47: 57-110.
Hilbert, David and Stephan CohnVossen. 1952. Geometry and the imagination. New York : Chelsea.
Sommerville, Duncan M. Y. 1958. An introduction to the geometry of N dimensions. New York: Dover Publications.
Stewart, Bonnie Madison. 1970. Adventures among the toroids. Michigan: Okemos.
Tietze, Heinrich. 1965. Famous problems of mathematics. New York: Greylock Press.
Thurston, William P. 1990. "Mathematical Education". Notices of A.M.S. 37, 7: 844-850.
Weil, Hermann. 1952. Symmetry. Princeton, New Jersey: Princeton University Press.
Wenninger, Magnus J. 1971. Polyhedron models. Cambridge: Cambridge University Press.
______. 1979. Spherical models. Cambridge: Cambridge University Press.
______. 1983. Dual models. Cambridge: Cambridge University Press.
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