MODERN ART AND THE HYPERCUBE ---------------------------- In the year 1925, writer André Breton penned the first Surrealist manifesto, a document which set out the basic principles of that artist movement. A declaration contained therein carried the signatures of the writers and artists Louis Aragon, Antonin Artaud, Jacques Baron, Joë Bousquet, J.-A. Boiffard, André Breton, Jean Carrive, René Crevel, Robert Desnos, Paul Élaurd, Max Ernst. Nowadays two artists, the Belgian René Magritte (1898-1967) and Spaniard Salvador Dali (1904-1989) probably rank as the painters most closely identified with Surrealism in the public mind . In particular, Dali's image of melting watches in his _The Persistence of Memory_ probably constitutes Surrealism's most recognizable icon. The young Dali certainly established himself as a ``bad boy'' of the art world and right up to the end he remained notorious for his relentless self-promotion. Nevertheless as an older man he flabbergasted his public by declaring himself a religious painter and turning to imagery derived from the most traditional art, but always with a difference. Consider his Cruxification (Corpus Hypercubicus) (1954). Dali offers a reasonably conventional depiction of Jesus on the cross, but the way he paints the cross itself defies all conventions. Indeed he renders the cross as a conglomeration of 8 cubes, a column of four with four more surrounding. (Four smaller cubes seem to float in the air in front of Jesus, but we shall say nothing more about these.) The meaning of the picture unfolds when one learns that the 8 cubes constitute a conventional 3-dimensional model for a 4-dimensional object known as a _hypercube_. Soon we shall turn to the mathematical development of the hypercube and its kindred structures and the whole idea of higher dimensions. For now, suffice to say that Dali aims for a sense of religious mystery transcending physical reality even as it descends to the physical world in the person of Jesus. He ingeniously conveys this using the metaphor of a hypercube, a 4-dimensional structure which itself transcends our physical world but which descends, by means of an unfolding process, to the realm of sensible perception. Dali counts only as one of many artists interested in representing the fourth dimension. Twenty years before the birth of Surrealism, Einstein proposed his special theory of relativity. (Don't take the word ``theory'' as suggesting anything tentative! Experiments have duly confirmed that Einstein's ideas describe the real world.) The special theory of relativity abolishes any shape distinction between space and time; the apparent distinction which we experience in everyday life emerges as an artifact of the comparitively small velocities of earthbound objects. Einstein argues that we can understand reality only by dealing with 4-dimensional space-time, that 3-dimensional space has no independent physical status, that words like ``now'' and ``the present'' play no part in an objective physical description of the universe. Needless to say, these radical proposals generated considerable excitement, and not just among scientists familiar with the technical details. Einstein's theories also captured the imagination of the general public, and the artists of the Modern movement could not ignore. Linda Dalrymple Henderson's book _The fourth dimension and non-Euclidean geometry in modern art_ goes into these matter in considerable detail. (Those students interested in the option of satisfying the requirements of this course by means of a substantial term paper might consider writing a careful review of its contents.) DIMENSION --------- Einstein certainly did not invent the idea of 4-dimensional space. His contribution consisted in elaborating the relevance of that idea to physics, but mathematicians had long before explored higher dimensional geometry for its own sake. For mathematicians, _dimension_ refers not to a single idea, but to a variety of related notions useful in diverse contexts. Therefore we shall not attempt here to give a single definition of dimension. Rather we shall concentrate on suggestive examples. Consider the following sequence of figures: .________. .________. | | |\ \ | | | \ \ | | | \ \ | | | \________\ | | | | | . ._______. | | | | | | | | | | .________. . | | Cube Point Line segment \ | | \ | | Square \ | | \._______. A POINT (tautologically) consists of a single point, and we need say nothing at all to specify it. A LINE SEGMENT contains infinitely many points, but we may specify on using a single real (=decimal) number. Indeed a mathematician makes little distinction between the geometrical notion of a line segment and the arithmetical notion of an interval of real numbers. A SQUARE also contains infinitely many points, but now specifying one such point requires two real numbers, two coordinates, say, a width and a height. As with the line segment, a mathematician makes little distinction between the geometrical notion of a square and the arithmetical notion of the set of all coordinate pairs drawn from a fixed interval. Specifying a point inside a CUBE requires three real numbers, say, a width, a height and a depth. Again, mathematicians usually conflate the geometrical notion of a CUBE with the arithmetical notion of the set of all coordinate triples drawn from a fixed interval. Now comes the interesting step. As far as our geometrical intuitions from daily life go, this sequence terminates immediately. But nothing prevents us from considering the set of all coordinate quadruples drawn from a fixed interval. From the arithmetic point of view, we can easily extend the sequence indefinitely. Not only can we extend the sequence, but we can generalize all the familiar geometrical notions which apply in the figures drawn above. We can specify which sets of quadruples of numbers count as lines, as planes, as (3-dimensional) spaces. We can define distance and angle and (hyper)volume, we can speak of symmetries, etc. Typically one works out all the details of a program like this in a LINEAR ALGEBRA course. Since strings of real numbers can describe many things besides the locations of points in a physical body, (the financial condition of business, the state of the global economy, the stresses on a bridge, the oscillations of an electronic circuit), reasoning about higher dimensions has many practical applications. POLYTOPES --------- We shall concentrate on one very special aspect of the fourth dimension, namely the analogs of the Platonic solids. In two dimensions on has an infinite sequence of geometrically distinct regular figures, the equilateral triangle, the square, the regular pentagon, regular hexagon, etc. In three dimensions one has exactly 5 regular polyhedra (precise definition below), the regular tetrahedron, cube, octahedron, dodecahedron and icosahedron. It turns out that in four dimensions one has 6 regular (we call them) polytopes. In five dimensions and higher, one has just three distinct types of polytopes. Perhaps this all comes as a surprise, one might well expect that everything grows more complicated as the dimensions increase. To define POLYTOPE we follow the pattern established by our familar polygons and polyhedra. A polygon (2 dimensional polytope, or 2-polytope for short) consists of line segments (1-polytopes) glued together in pairs at their endpoints (0-polytopes). A polyhedron (3-polytope) consists of polygons (2-polytopes) glued together in pairs along their edges (1-polytopes). A 4-polytope consists of polyhedra (3-polytopes, we'll call them BODIES) glued together in pairs along their faces (2-polytopes). . . . A n-polytope consists of (n-1)-polytopes glued together in pairs along their (n-2)-polytopes. Since 4-polytopes will keep us plenty busy, we won't have much to say about dimensions higher than that. Of course we find 4-polytopes difficult to picture directly, so we shall mostly deal with them as collections of polyhedra with gluing instructions. One can also deal with a higher dimensional object by means of its ``shadows.'' After all, we habitually use 2-dimensional retinas to see the 3-dimensional world. 4-polytopes also have (usually very complicated) 2-dimensional shadows. Even better, we can use stereo-optical pictures to have a look at their 3-dimensional ``shadows.'' That will come very close to giving us a direct experience of 4-dimensional exotica. THE LINK OF A VERTEX IN A POLYTOPE ---------------------------------- If you take a sharp knife and slice of a vertex of a cube, you will see a triangular cross-section. If you slice of a vertex of an octahedron, you will see a square cross-section. We shall call these cross-sectional polygons the LINK of the that vertex. We call a polyhedron regular if (i) it has regular polygon faces, all of the same shape; (ii) it has regular polygon links, all of the same shape. (The links and faces need not have the same shape.) Let's give a slightly more formal description of the concept of a link: The _vertices_ of _the link at vertex V_ correspond to those edges of the polyhedron which contain V. The _edges_ of _the link at vertex V_ correspond to those faces of the polyhedron which contain V. An _edge_ of the link connects two _vertices_ of the link if the corresponding _face_ contains both the corresponding edges. You should carefully read over this formal definition, and compare it to the informal treatment via ``slicing.'' Seeing how these two match help helps greatly when we pass on to 4-dimensions. Now lets give a formal description of a link of a vertex in a 4-polytope: The _vertices_ of _the link at vertex v_ correspond to those edges of the polytope which contain V. The _edges_ of _the link at vertex v_ correspond to those faces of the polytope which contain V. The _faces_ of _the link at vertex v_ correspond to those bodies of the polytope which contain V. A _face_ of the link contains an _edge_ of the link if the corresponding body contains the corresponding face. An _edge_ of the link contains an _vertex_ of the link if the corresponding face contains the corresponding edge. Finally, we shall call a 4-polytope regular if (i) it has regular polyhedral bodies, all of the same shape; (ii) it has regular polyhedral links, all of the same shape. As before, the bodies and the links need not have the same shape. We end this section with simple, but CRUCIAL observation. Inside a 4-polytope p, fix your consideration a particular vertex v of a particular body b . Then THE LINK OF b AT VERTEX v equals THE FACE OF THE LINK OF p AT v WHICH CORRESPONDS TO b. Basically, this says that when you slice of a corner of a 4-polytope, you also slice of corners of all the bodies that meet there. The polygons which form the cross-sections of the bodies constitute the faces of link.