Alternation (geometry)
In geometry, an alternation or partial truncation, is an operation on a polygon, tiling, or higher dimensional polytope that removes alternate vertices. Coxeter labels an alternation by a prefixed h, standing for half; because alternation reduces all polygon faces to half as many sides, it can only be applied to polytopes with all even-sided faces. An alternated square face becomes a digon, being degenerate, is reduced to a single edge. More any vertex-uniform polyhedron or tiling with a vertex configuration consisting of all even-numbered elements can be alternated. For example, the alternation of a vertex figure with 2a.2b.2c is a.3.b.3.c.3 where the three is the number of elements in this vertex figure. A special case is square faces. So for example, the cube 4.4.4 is alternated as 2.3.2.3.2.3, reduced to 3.3.3, being the tetrahedron, all the 6 edges of the tetrahedra can be seen as the degenerate faces of the original cube. A snub can be seen as an alternation of a truncated truncated quasiregular polyhedron.
In general a polyhedron can be snubbed. All truncated rectified polyhedra can be snubbed, not just from regular polyhedra; the snub square antiprism is an example of a general snub, can be represented by ss, with the square antiprism, s. This alternation operation applies to higher-dimensional polytopes and honeycombs as well, but in general most of the results of this operation will not be uniform; the voids created by the deleted vertices will not in general create uniform facets, there are not enough degrees of freedom to allow an appropriate rescaling of the new edges. Exceptions do exist, such as the derivation of the snub 24-cell from the truncated 24-cell. Examples: Honeycombs An alternated cubic honeycomb is the tetrahedral-octahedral honeycomb. An alternated hexagonal prismatic honeycomb is the gyrated alternated cubic honeycomb. 4-polytope An alternated truncated 24-cell is the snub 24-cell. 4-honeycombs: An alternated truncated 24-cell honeycomb is the snub 24-cell honeycomb. A hypercube can always be alternated into a uniform demihypercube.
Cube → Tetrahedron → Tesseract → 16-cell → Penteract → demipenteract Hexeract → demihexeract... Coxeter used the operator a, which contains both halves, so retains the original symmetry. For even-sided regular polyhedra, a represents a compound polyhedron with two opposite copies of h. For odd-sided, greater than 3, regular polyhedra a, becomes a star polyhedron. Norman Johnson extended the use of the altered operator a, b for blended, c for converted, as, respectively; the compound polyhedron, stellated octahedron can be represented by and. The star-polyhedron, small ditrigonal icosidodecahedron, can be represented by and. Here all the pentagons have been alternated into pentagrams, triangles have been inserted to take up the resulting free edges. A similar operation can truncate alternate vertices, rather than just removing them. Below is a set of polyhedra; these have two types of vertices. Truncating the "higher order" vertices and both vertex types produce these forms: Conway polyhedral notation Wythoff construction Coxeter, H.
S. M. Regular Polytopes, Dover edition, ISBN 0-486-61480-8 Norman Johnson Uniform Polytopes, Manuscript N. W. Johnson: The Theory of Uniform Polytopes and Honeycombs, Ph. D. Dissertation, University of Toronto, 1966 Weisstein, Eric W. "Snubification". MathWorld. Richard Klitzing, alternated facetings, Stott-Coxeter-Dynkin diagrams, Symmetry: Culture and Science, Vol. 21, No.4, 329-344, George. "Alternation". Glossary for Hyperspace. Archived from the original on 4 February 2007. Polyhedra Names, snub
Hexagonal tiling
In geometry, the hexagonal tiling or hexagonal tessellation is a regular tiling of the Euclidean plane, in which three hexagons meet at each vertex. It has Schläfli symbol of or t. English mathematician John Conway calls it a hextille; the internal angle of the hexagon is 120 degrees so three hexagons at a point make a full 360 degrees. It is one of three regular tilings of the plane; the other two are the square tiling. The hexagonal tiling is the densest way to arrange circles in two dimensions; the Honeycomb conjecture states that the hexagonal tiling is the best way to divide a surface into regions of equal area with the least total perimeter. The optimal three-dimensional structure for making honeycomb was investigated by Lord Kelvin, who believed that the Kelvin structure is optimal. However, the less regular Weaire–Phelan structure is better; this structure exists in the form of graphite, where each sheet of graphene resembles chicken wire, with strong covalent carbon bonds. Tubular graphene sheets have been synthesised.
They have many potential applications, due to electrical properties. Silicene is similar. Chicken wire consists of a hexagonal lattice of wires; the hexagonal tiling appears in many crystals. In three dimensions, the face-centered cubic and hexagonal close packing are common crystal structures, they are the densest known sphere packings in three dimensions, are believed to be optimal. Structurally, they comprise parallel layers of hexagonal tilings, similar to the structure of graphite, they differ in the way that the layers are staggered from each other, with the face-centered cubic being the more regular of the two. Pure copper, amongst other materials, forms a face-centered cubic lattice. There are three distinct uniform colorings of a hexagonal tiling, all generated from reflective symmetry of Wythoff constructions; the represent the periodic repeat of one colored tile, counting hexagonal distances as h first, k second. The same counting is used in the Goldberg polyhedra, with a notation h,k, can be applied to hyperbolic tilings for p>6.
The 3-color tiling is a tessellation generated by the order-3 permutohedrons. A chamferred hexagonal tiling replacing edges with new hexagons and transforms into another hexagonal tiling. In the limit, the original faces disappear, the new hexagons degenerate into rhombi, it becomes a rhombic tiling; the hexagons can be dissected into sets of 6 triangles. This process leads to two 2-uniform tilings, the triangular tiling: The hexagonal tiling can be considered an elongated rhombic tiling, where each vertex of the rhombic tiling is stretched into a new edge; this is similar to the relation of the rhombic dodecahedron and the rhombo-hexagonal dodecahedron tessellations in 3 dimensions. It is possible to subdivide the prototiles of certain hexagonal tilings by two, four or nine equal pentagons: This tiling is topologically related as a part of sequence of regular tilings with hexagonal faces, starting with the hexagonal tiling, with Schläfli symbol, Coxeter diagram, progressing to infinity; this tiling is topologically related to regular polyhedra with vertex figure n3, as a part of sequence that continues into the hyperbolic plane.
It is related to the uniform truncated polyhedra with vertex figure n.6.6. This tiling is a part of a sequence of truncated rhombic polyhedra and tilings with Coxeter group symmetry; the cube can be seen as a rhombic hexahedron. The truncated forms have regular n-gons at the truncated vertices, nonregular hexagonal faces. Like the uniform polyhedra there are eight uniform tilings that can be based from the regular hexagonal tiling. Drawing the tiles colored as red on the original faces, yellow at the original vertices, blue along the original edges, there are 8 forms, 7 which are topologically distinct. There are 3 types of monohedral convex hexagonal tilings, they are all isohedral. Each has parametric variations within a fixed symmetry. Type 2 contains glide reflections, is 2-isohedral keeping chiral pairs distinct. Hexagonal tilings can be made with the identical topology as the regular tiling. With isohedral faces, there are 13 variations. Symmetry given assumes all faces are the same color. Colors here represent the lattice positions.
Single-color lattices are parallelogon hexagons. Other isohedrally-tiled topological hexagonal tilings are seen as quadrilaterals and pentagons that are not edge-to-edge, but interpreted as colinear adjacent edges: The 2-uniform and 3-uniform tessellations have a rotational degree of freedom which distorts 2/3 of the hexagons, including a colinear case that can be seen as a non-edge-to-edge tiling of hexagons and larger triangles, it can be distorted into a chiral 4-colored tri-directional weaved pattern, distorting some hexagons into parallelograms. The weaved pattern with 2 colored faces have rotational 632 symmetry. A chevron pattern has pmg symmetry, lowered to p1 with 3 or 4 colored tiles; the hexagonal tiling can be used as a circle packing, placing equal diameter circles at the center of every point. Every circle is in contact with 3 other circles in the packing; the gap inside each hexagon allows for one circle, creating the densest packing from the triangular tiling, with each circle contact with the maximum of 6 circles.
There are 2 regular complex apeirogons, sharing the vertices of the
Catalan solid
In mathematics, a Catalan solid, or Archimedean dual, is a dual polyhedron to an Archimedean solid. There are 13 Catalan solids, they are named for the Belgian mathematician, Eugène Catalan, who first described them in 1865. The Catalan solids are all convex, they are face-transitive but not vertex-transitive. This is because the dual Archimedean solids are not face-transitive. Note that unlike Platonic solids and Archimedean solids, the faces of Catalan solids are not regular polygons. However, the vertex figures of Catalan solids are regular, they have constant dihedral angles. Being face-transitive, Catalan solids are isohedra. Additionally, two of the Catalan solids are edge-transitive: the rhombic dodecahedron and the rhombic triacontahedron; these are the duals of the two quasi-regular Archimedean solids. Just as prisms and antiprisms are not considered Archimedean solids, so bipyramids and trapezohedra are not considered Catalan solids, despite being face-transitive. Two of the Catalan solids are chiral: the pentagonal icositetrahedron and the pentagonal hexecontahedron, dual to the chiral snub cube and snub dodecahedron.
These each come in two enantiomorphs. Not counting the enantiomorphs and trapezohedra, there are a total of 13 Catalan solids; the Catalan solids, along with their dual Archimedean solids, can be grouped in those with tetrahedral and icosahedral symmetry. For both octahedral and icosahedral symmetry there are six forms; the only Catalan solid with genuine tetrahedral symmetry is the triakis tetrahedron. Rhombic dodecahedron and tetrakis hexahedron have octahedral symmetry, but they can be colored to have only tetrahedral symmetry. Rectification and snub exist with tetrahedral symmetry, but they are Platonic instead of Archimedean, so their duals are Platonic instead of Catalan. List of uniform tilings Shows dual uniform polygonal tilings similar to the Catalan solids Conway polyhedron notation A notational construction process Archimedean solid Johnson solid Eugène Catalan Mémoire sur la Théorie des Polyèdres. J. l'École Polytechnique 41, 1-71, 1865. Alan Holden Shapes and Symmetry. New York: Dover, 1991.
Wenninger, Dual Models, Cambridge University Press, ISBN 978-0-521-54325-5, MR 0730208 Williams, Robert. The Geometrical Foundation of Natural Structure: A Source Book of Design. Dover Publications, Inc. ISBN 0-486-23729-X. Anthony Pugh. Polyhedra: A visual approach. California: University of California Press Berkeley. ISBN 0-520-03056-7. Chapter 4: Duals of the Archimedean polyhedra and antiprisms Weisstein, Eric W. "Catalan Solids". MathWorld. Weisstein, Eric W. "Isohedron". MathWorld. Archimedean duals – at Virtual Reality Polyhedra Interactive Catalan Solid in Java
Dual polyhedron
In geometry, any polyhedron is associated with a second dual figure, where the vertices of one correspond to the faces of the other and the edges between pairs of vertices of one correspond to the edges between pairs of faces of the other. Such dual figures remain combinatorial or abstract polyhedra, but not all are geometric polyhedra. Starting with any given polyhedron, the dual of its dual is the original polyhedron. Duality preserves the symmetries of a polyhedron. Therefore, for many classes of polyhedra defined by their symmetries, the duals belong to a symmetric class. Thus, the regular polyhedra – the Platonic solids and Kepler–Poinsot polyhedra – form dual pairs, where the regular tetrahedron is self-dual; the dual of an isogonal polyhedron, having equivalent vertices, is one, isohedral, having equivalent faces. The dual of an isotoxal polyhedron is isotoxal. Duality is related to reciprocity or polarity, a geometric transformation that, when applied to a convex polyhedron, realizes the dual polyhedron as another convex polyhedron.
There are many kinds of duality. The kinds most relevant to elementary polyhedra are polar reciprocity and topological or abstract duality; the duality of polyhedra is defined in terms of polar reciprocation about a concentric sphere. Here, each vertex is associated with a face plane so that the ray from the center to the vertex is perpendicular to the plane, the product of the distances from the center to each is equal to the square of the radius. In coordinates, for reciprocation about the sphere x 2 + y 2 + z 2 = r 2, the vertex is associated with the plane x 0 x + y 0 y + z 0 z = r 2; the vertices of the dual are the poles reciprocal to the face planes of the original, the faces of the dual lie in the polars reciprocal to the vertices of the original. Any two adjacent vertices define an edge, these will reciprocate to two adjacent faces which intersect to define an edge of the dual; this dual pair of edges are always orthogonal to each other. If r 0 is the radius of the sphere, r 1 and r 2 the distances from its centre to the pole and its polar, then: r 1.
R 2 = r 0 2 For the more symmetrical polyhedra having an obvious centroid, it is common to make the polyhedron and sphere concentric, as in the Dorman Luke construction described below. However, it is possible to reciprocate a polyhedron about any sphere, the resulting form of the dual will depend on the size and position of the sphere; the choice of center for the sphere is sufficient to define the dual up to similarity. If multiple symmetry axes are present, they will intersect at a single point, this is taken to be the centroid. Failing that, a circumscribed sphere, inscribed sphere, or midsphere is used. If a polyhedron in Euclidean space has an element passing through the center of the sphere, the corresponding element of its dual will go to infinity. Since Euclidean space never reaches infinity, the projective equivalent, called extended Euclidean space, may be formed by adding the required'plane at infinity'; some theorists prefer to say that there is no dual. Meanwhile, Wenninger found a way to represent these infinite duals, in a manner suitable for making models.
The concept of duality here is related to the duality in projective geometry, where lines and edges are interchanged. Projective polarity works well enough for convex polyhedra, but for non-convex figures such as star polyhedra, when we seek to rigorously define this form of polyhedral duality in terms of projective polarity, various problems appear. Because of the definitional issues for geometric duality of non-convex polyhedra, Grünbaum argues that any proper definition of a non-convex polyhedron should include a notion of a dual polyhedron. Any convex polyhedron can be distorted into a canonical form, in which a unit midsphere exists tangent to every edge, such that the average position of the points of tangency is the center of the sphere; this form is unique up to congruences. If we reciprocate such a canonical polyhedron about its midsphere, the dual polyhedron will share the same edge-tangency points and so must be canonical, it is the canonical dual, the two together form a canonical dual pair.
When a pair of polyhedra cannot be obtained by reciprocation from each other, they may be called duals of each other as long as the vertices of one correspond to the faces of the other, the edges of one correspond to the edges of the other, in an incidence-preserving way. Such pairs of polyhedra are abstractly dual; the vertices and edges of a convex polyhedron form a graph, embedded on a topological sphere, the surface of the polyhedron. The same graph can be projected to form
Dual graph
In the mathematical discipline of graph theory, the dual graph of a plane graph G is a graph that has a vertex for each face of G. The dual graph has an edge whenever two faces of G are separated from each other by an edge, a self-loop when the same face appears on both sides of an edge. Thus, each edge e of G has a corresponding dual edge, whose endpoints are the dual vertices corresponding to the faces on either side of e; the definition of the dual depends on the choice of embedding of the graph G, so it is a property of plane graphs rather than planar graphs. For planar graphs there may be multiple dual graphs, depending on the choice of planar embedding of the graph; the first form of graph duality to be recognized was the association of the Platonic solids into pairs of dual polyhedra. Graph duality is a topological generalization of the geometric concepts of dual polyhedra and dual tessellations, is in turn generalized algebraically by the concept of a dual matroid. Variations of planar graph duality include a version of duality for directed graphs, duality for graphs embedded onto non-planar two-dimensional surfaces.
However, these notions of dual graphs should not be confused with a different notion, the edge-to-vertex dual or line graph of a graph. The term "dual" is used because the property of being a dual graph is symmetric, meaning that if H is a dual of a connected graph G G is a dual of H; when discussing the dual of a graph G, the graph G itself may be referred to as the "primal graph". Many other graph properties and structures may be translated into other natural properties and structures of the dual. For instance, cycles are dual to cuts, spanning trees are dual to the complements of spanning trees, simple graphs are dual to 3-edge-connected graphs. Graph duality can help explain the structure of mazes and of drainage basins. Dual graphs have been applied in computer vision, computational geometry, mesh generation, the design of integrated circuits; the unique planar embedding of a cycle graph divides the plane into only two regions, the inside and outside of the cycle, by the Jordan curve theorem.
However, in an n-cycle, these two regions are separated from each other by n different edges. Therefore, the dual graph of the n-cycle is a multigraph with two vertices, connected to each other by n dual edges; such a graph is called a dipole graph. Conversely, the dual to an n-edge dipole graph is an n-cycle. According to Steinitz's theorem, every polyhedral graph must be planar and 3-vertex-connected, every 3-vertex-connected planar graph comes from a convex polyhedron in this way; every three-dimensional convex polyhedron has a dual polyhedron. Whenever two polyhedra are dual, their graphs are dual. For instance the Platonic solids come in dual pairs, with the octahedron dual to the cube, the dodecahedron dual to the icosahedron, the tetrahedron dual to itself. Polyhedron duality can be extended to duality of higher dimensional polytopes, but this extension of geometric duality does not have clear connections to graph-theoretic duality. A plane graph is said to be self-dual; the wheel graphs provide an infinite family of self-dual graphs coming from self-dual polyhedra.
However, there exist self-dual graphs that are not polyhedral, such as the one shown. Servatius & Christopher describe two operations and explosion, that can be used to construct a self-dual graph containing a given planar graph, it follows from Euler's formula that every self-dual graph with n vertices has 2n − 2 edges. Every simple self-dual planar graph contains at least four vertices of degree three, every self-dual embedding has at least four triangular faces. Many natural and important concepts in graph theory correspond to other natural but different concepts in the dual graph; because the dual of the dual of a connected plane graph is isomorphic to the primal graph, each of these pairings is bidirectional: if concept X in a planar graph corresponds to concept Y in the dual graph concept Y in a planar graph corresponds to concept X in the dual. The dual of a simple graph need not be simple: it may have self-loops or multiple edges connecting the same two vertices, as was evident in the example of dipole multigraphs being dual to cycle graphs.
As a special case of the cut-cycle duality discussed below, the bridges of a planar graph G are in one-to-one correspondence with the self-loops of the dual graph. For the same reason, a pair of parallel edges in a dual multigraph corresponds to a 2-edge cutset in the primal graph. Therefore, a planar graph is only if its dual has no 1 - or 2-edge cutsets; the simple planar graphs whose duals are simple are the 3-edge-connected simple planar graphs. This class of graphs includes, but is not the same as, the class of 3-vertex-connected simple planar graphs. For instance, the figure showing a self-dual graph is 3-edge-connected but is not 3-vertex-connected; because the dual graph depends on a particular embedding, the dual graph of a p
Wythoff construction
In geometry, a Wythoff construction, named after mathematician Willem Abraham Wythoff, is a method for constructing a uniform polyhedron or plane tiling. It is referred to as Wythoff's kaleidoscopic construction; the method is based on the idea of tiling a sphere, with spherical triangles – see Schwarz triangles. This construction arranges three mirrors at the sides of a triangle, like in a kaleidoscope. However, different from a kaleidoscope, the mirrors are not parallel, but intersect at a single point, they therefore enclose a spherical triangle on the surface of any sphere centered on that point and repeated reflections produce a multitude of copies of the triangle. If the angles of the spherical triangle are chosen appropriately, the triangles will tile the sphere, one or more times. If one places a vertex at a suitable point inside the spherical triangle enclosed by the mirrors, it is possible to ensure that the reflections of that point produce a uniform polyhedron. For a spherical triangle ABC we have four possibilities which will produce a uniform polyhedron: A vertex is placed at the point A.
This produces a polyhedron with Wythoff symbol a|b c, where a equals π divided by the angle of the triangle at A, for b and c. A vertex is placed at a point on line AB so that it bisects the angle at C; this produces a polyhedron with Wythoff symbol a b|c. A vertex is placed so that it is on the incenter of ABC; this produces a polyhedron with Wythoff symbol a b c|. The vertex is at a point such that, when it is rotated around any of the triangle's corners by twice the angle at that point, it is displaced by the same distance for every angle. Only even-numbered reflections of the original vertex are used; the polyhedron has the Wythoff symbol |a b c. The process in general applies for higher-dimensional regular polytopes, including the 4-dimensional uniform 4-polytopes. Uniform polytopes that cannot be created through a Wythoff mirror construction are called non-Wythoffian, they can be derived from Wythoffian forms either by alternation or by insertion of alternating layers of partial figures. Both of these types of figures will contain rotational symmetry.
Sometimes snub forms are considered Wythoffian though they can only be constructed by the alternation of omnitruncated forms. Wythoff symbol - a symbol for the Wythoff construction of uniform polyhedra and uniform tilings. Coxeter-Dynkin diagram - a generalized symbol for the Wythoff construction of uniform polytopes and honeycombs. Coxeter Regular Polytopes, Third edition, Dover edition, ISBN 0-486-61480-8 Coxeter The Beauty of Geometry: Twelve Essays, Dover Publications, 1999, ISBN 0-486-40919-8 Har'El, Z. Uniform Solution for Uniform Polyhedra. Geometriae Dedicata 47, 57-110, 1993. W. A. Wythoff, A relation between the polytopes of the C600-family, Koninklijke Akademie van Wetenschappen te Amsterdam, Proceedings of the Section of Sciences, 20 966–970. Greg Egan's applet to display uniform polyhedra using Wythoff's construction method A Shadertoy renderization of Wythoff's construction methodJenn, software that generates views of polyhedra and polychora from symmetry groups
Euclidean geometry
Euclidean geometry is a mathematical system attributed to Alexandrian Greek mathematician Euclid, which he described in his textbook on geometry: the Elements. Euclid's method consists in assuming a small set of intuitively appealing axioms, deducing many other propositions from these. Although many of Euclid's results had been stated by earlier mathematicians, Euclid was the first to show how these propositions could fit into a comprehensive deductive and logical system; the Elements begins with plane geometry, still taught in secondary school as the first axiomatic system and the first examples of formal proof. It goes on to the solid geometry of three dimensions. Much of the Elements states results of what are now called algebra and number theory, explained in geometrical language. For more than two thousand years, the adjective "Euclidean" was unnecessary because no other sort of geometry had been conceived. Euclid's axioms seemed so intuitively obvious that any theorem proved from them was deemed true in an absolute metaphysical, sense.
Today, many other self-consistent non-Euclidean geometries are known, the first ones having been discovered in the early 19th century. An implication of Albert Einstein's theory of general relativity is that physical space itself is not Euclidean, Euclidean space is a good approximation for it only over short distances. Euclidean geometry is an example of synthetic geometry, in that it proceeds logically from axioms describing basic properties of geometric objects such as points and lines, to propositions about those objects, all without the use of coordinates to specify those objects; this is in contrast to analytic geometry, which uses coordinates to translate geometric propositions into algebraic formulas. The Elements is a systematization of earlier knowledge of geometry, its improvement over earlier treatments was recognized, with the result that there was little interest in preserving the earlier ones, they are now nearly all lost. There are 13 books in the Elements: Books I–IV and VI discuss plane geometry.
Many results about plane figures are proved, for example "In any triangle two angles taken together in any manner are less than two right angles." and the Pythagorean theorem "In right angled triangles the square on the side subtending the right angle is equal to the squares on the sides containing the right angle." Books V and VII–X deal with number theory, with numbers treated geometrically as lengths of line segments or areas of regions. Notions such as prime numbers and rational and irrational numbers are introduced, it is proved. Books XI–XIII concern solid geometry. A typical result is the 1:3 ratio between the volume of a cone and a cylinder with the same height and base; the platonic solids are constructed. Euclidean geometry is an axiomatic system, in which all theorems are derived from a small number of simple axioms; until the advent of non-Euclidean geometry, these axioms were considered to be true in the physical world, so that all the theorems would be true. However, Euclid's reasoning from assumptions to conclusions remains valid independent of their physical reality.
Near the beginning of the first book of the Elements, Euclid gives five postulates for plane geometry, stated in terms of constructions: Let the following be postulated:To draw a straight line from any point to any point. To produce a finite straight line continuously in a straight line. To describe a circle with any centre and distance; that all right angles are equal to one another.: That, if a straight line falling on two straight lines make the interior angles on the same side less than two right angles, the two straight lines, if produced indefinitely, meet on that side on which the angles are less than two right angles. Although Euclid only explicitly asserts the existence of the constructed objects, in his reasoning they are implicitly assumed to be unique; the Elements include the following five "common notions": Things that are equal to the same thing are equal to one another. If equals are added to equals the wholes are equal. If equals are subtracted from equals the differences are equal.
Things that coincide with one another are equal to one another. The whole is greater than the part. Modern scholars agree that Euclid's postulates do not provide the complete logical foundation that Euclid required for his presentation. Modern treatments use more complete sets of axioms. To the ancients, the parallel postulate seemed less obvious than the others, they aspired to create a system of certain propositions, to them it seemed as if the parallel line postulate required proof from simpler statements. It is now known that such a proof is impossible, since one can construct consistent systems of geometry in which the parallel postulate is true, others in which it is false. Euclid himself seems to have considered it as being qualitatively different from the others, as evidenced by the organization of the Elements: his first 28 propositions are those that can be proved without it. Many alternative axioms can be formulated. For example, Playfair's axiom states: In a plane, through a point not on a given straight line, at most one line can be drawn that never meets the giv