Add another dimension to your life: with a bonus recipe for making tesseracts.Let me pose a simple question: What is Figure 1? [FIGURE 1 OMITTED] Did you answer "A cube cube, in geometry, regular solid bounded by six equal squares. All adjacent faces of a cube are perpendicular to each other; any one face of a cube may be its base. The dimensions of a cube are the lengths of the three edges which meet at any vertex. "? Figure 1 certainly looks like a cube, and I suspect a good many of you identified it as such, but it is not actually a cube. A cube is a solid in three dimensions, with three mutually perpendicular right angles evident at the vertices The plural of vertex. See vertex. . The thing in Figure 1 is not a solid: it is flat, and so it is not a cube. In fact, Figure 1 is only a two-dimensional representation of a cube, since a cube's three-dimensional nature cannot exist in the two-dimensional plane of a piece of paper. In drawing the cube in two dimensions (which is what we have done in Figure 1), we have to cheat and draw one of the right angles at less than 90 degrees. This insistence that Figure 1 is not a cube but merely a representation of one may seem a little pedantic pe·dan·tic adj. Characterized by a narrow, often ostentatious concern for book learning and formal rules: a pedantic attention to details. yet it is critical for understanding about shapes, and for allowing us to try to imagine the fourth dimension. Understanding the space in which we live Before we go in search of this extra dimension, let us review what we know about zero, one, two and three dimensions, and certain shapes in each of these. We will start with the first three cases, because we can draw things realistically for these. So, imagine a dot--with no width, no breadth, and no depth (which means that my picture of it in Figure 2 is actually not very realistic at all!). This is the 0-dimensional universe, where you have no place to go but where you already are! Here there are no directions: no ups nor downs, no lefts nor rights, no backs nor forths. [FIGURE 2 OMITTED] Now imagine that we can take the point and move it in one direction only, say horizontally from left to right across the page. If we allow the dot to move infinitely far to the left and the right, we will create an infinite straight line. This is the one-dimensional universe, where you have some freedom to move provided you only want to go left and right along the line. If we mark a point as the origin, then a single number or co-ordinate is enough to tell us where we are in relationship to that origin (this co-ordinate tells us how far to the left or right of the origin we are). Although we can now move, we still cannot go up and down, nor backwards and forwards. If, instead of allowing the dot to move an infinite distance in·fi·nite distance n. A distance of 20 feet or more, at which light rays entering the eyes are practically parallel. , I allow the original dot to move only 3 cm to the right, as suggested in Figure 2 (imagining that the line has no thickness), then I end up with a one-dimensional line segment that is 3 cm long. This segment is just a part of the whole one-dimensional universe, but it is a very important shape in its own right. Now, let us take our one-dimensional line and imagine that we can move it at right angles so as to form a right angle or right angles, as when one line crosses another perpendicularly. See also: Right to itself. Since we have used horizontally left and right across the page as one direction, we need to go vertically up and down the page. The result is a two-dimensional plane, extending infinitely left-right and up-down. By moving various amounts left and right, and up and down, I can get anywhere in this two-dimensional universe, but I am stuck in the plane and cannot move backwards and forwards (or inwards in·ward adj. 1. Located inside; inner. 2. Directed or moving toward the interior: an inward flow. 3. and outwards out·ward adj. 1. Of, located on, or moving toward the outside or exterior; outer. 2. Relating to the physical self: a concern with outward beauty rather than with inward reflections. ). In two-dimensional space, once I pick a point as the origin, it takes two coordinates to describe the position of another point: how far to the left/right of the origin it is, and how far up/down. If we take only the 3 cm one-dimensional line segment and move it at right angles up the page for just 3 cm, then we get a square. This is illustrated in Figure 2. The square is two-dimensional object--a subset A group of commands or functions that do not include all the capabilities of the original specification. Software or hardware components designed for the subset will also work with the original. of all of two-dimensional space--and includes both the interior and the boundary. In order to illustrate better the rest of our discussion, however, we are going to highlight the edges of the square, and make the interior transparent as shown in the first part of Figure 3. [FIGURE 3 OMITTED] Things now start getting a little tricky Little Tricky was a horse ridden by American Bruce Davidson in the sport of eventing.
In geometry, a two- or three-dimensional space in which the axioms and postulates of Euclidean geometry apply; also, a space in any finite number of dimensions, in which points are designated by coordinates (one for each dimension) and the distance between . Now there is freedom to move up-down, left-right, and backwards-forwards, and the position of every point can be described in relation to an origin by three co-ordinates. Let us return to our cube, and the way I got you to imagine its construction, moving the square away from your face. This is all well and good for you in three-dimensional space Three-dimensional space is the physical universe we live in. The three dimensions are commonly called length, width, and breadth, although any three mutually perpendicular directions can serve as the three dimensions. Pictures are commonly two dimensional, they lack depth. , where you can actually move the square away from you. For me writing this article, however, I have a problem in trying to show this. I have to represent that third dimension--the one that goes into and out of the page--by a sloping line. So, I move my square upwards and to the right on the page, in order to try and show a third direction that is mutually perpendicular to the other two. This third right angle ends up not being a right angle at all, but it is the best we can do on a flat piece of paper. This approach gives us the quasi-perspective representation shown in Figure 3--the standard drawing of a cube--with sloping lines and some of the faces looking like parallelograms. In fact, it should be noted that the way that people often draw such a cube is to draw two squares, offset from one another, and then join the corners with sloping lines. Before we go any further, I want you to compare your "real" cube (the one you pictured by moving that square away from your nose) with my drawing in Figure 3. There is much that is missing from my drawing. For starters there is an inside space within the real cube, where you can look out through six faces by looking up, down, left, right, forwards and backwards. My drawing has no inside space. The faces that you see from inside the real cube are all square; my drawing has only two faces that actually look like squares whereas the rest look like parallelograms. At every corner of the real cube there are three edges that meet as a set of three mutually perpendicular lines (i.e., each one is at right angles to each of the others). On my diagram there are three lines meeting, but very few of them are shown to meet as actual right angles. Above all, my diagram has no depth: that third dimension is missing. Thinking outside the square/cube Let us review what we have done so far, and think about it more generally. At each stage we have taken an n-dimensional object and turned it into an (n+1)-dimensional object by moving it in the direction of a new right angle. This allowed us to go from a point to a line segment to a square and finally to a cube. In the case of going from two to three dimensions, the third right angle can be shown correctly provided you are working in three-dimensional space; but, if working on paper, it is difficult to show that third direction and capture all the features that characterise Verb 1. characterise - be characteristic of; "What characterizes a Venetian painting?" characterize differentiate, distinguish, mark - be a distinctive feature, attribute, or trait; sometimes in a very positive sense; "His modesty distinguishes him from his a three-dimensional cube. We are going to extend these ideas to see if we can at least start to visualise four dimensions. I want you to imagine a cube, a real three-dimensional cube sitting in space. Now take this cube and mentally move it at right angles to all the other right angles that you have, and move it a distance equal to the lengths of the edges of the cube. (Just like we did with the square, to make it a cube.) "But wait," I hear you say. "There's no such right angle." Okay, you are right; there is only room for three right angles in three-dimensional space. So let us cheat--the way I had to cheat when I drew a three-dimensional cube on a two-dimensional piece of paper--by drawing the extra right angle at something other than 90 degrees. Consequently, imagine the cube, in space, and move it along a sloping line for a distance equal to the cube's side length. It might help if I give you a recipe for building what I am trying to describe: instructions for making a (three-dimensional representation of a) tesseract or hypercube A parallel processing architecture made up of binary multiples of computers (4, 8, 16, etc.). The computers are interconnected so that data travel is kept to a minimum. For example, in two eight-node cubes, each node in one cube would be connected to the counterpart node in the other. , or the four-dimensional equivalent of a cube. "Hypercube" is the more general term and applies in dimensions higher than three; the word tesseract is usually reserved for the specific four-dimensional case. A recipe for the fourth dimension Ingredients 32 plastic drinking straws (preferably of the same bright colour) A reel of cotton 8 medium-sized blobs of Blu-tack Needle Magnet magnet: see electromagnet; magnetism. magnet Any material capable of attracting iron and producing a magnetic field outside itself. By the end of the 19th century, all known elements and many compounds had been tested for magnetism, and all were Easily accessible ceiling or a box (half-straws in a photocopy paper box work quite well) Method The model of a tesseract can be constructed by threading the cotton through the straws. It is best to use 1-2 metre lengths of cotton, and when one length is nearly used up tie on a new length and continue threading. Furthermore, it is easier to build the tesseract in its final location, as hyper-cubes are not easily transportable (particularly in three dimensions!). Consequently, make sure you have a comfortable platform from which to work. Attach the cotton firmly to one end of a straw, and then thread the cotton through this straw and three others, before passing the cotton through the first straw to make a square. Keep the square taut taut adj. taut·er, taut·est 1. Pulled or drawn tight; not slack. See Synonyms at tight. 2. Strained; tense: nerves taut with anxiety. 3. a. (or adjust it as you go) and use four lengths of cotton and the Blu-tack to suspend the square from the ceiling, or from the inside top of a large box, as shown in Figure 4. [FIGURE 4 OMITTED] Use this square as the top face of a cube which you now build using another eight straws. The cube hangs down from the top face, and is constructed by threading the cotton through both the new straws and those already in position, with the magnet being used to draw the needle and thread through the straws. It is necessary to keep pulling the cotton taut--if you do this you will not have to tie any knots, although you may like to for added security. While threading the cube you may like to contemplate if there is a minimum length of thread to use! However, use as much as is needed to make everything tight. When you have finished building the cube, tie off the end of the cotton so that the tension does not ease. You now have a three-dimensional cube suspended sus·pend v. sus·pend·ed, sus·pend·ing, sus·pends v.tr. 1. To bar for a period from a privilege, office, or position, usually as a punishment: suspend a student from school. in three-dimensional space, where in this case we added the third dimension to the original square by coming downwards. From here the job becomes a little more delicate--and a lot more mind-blowing. Thread four straws onto a new piece of cotton, but before joining these into a square locate them as shown in Figure 5, so that the new square is interlocked with the first cube. Suspend this square from the ceiling with four pieces of cotton and Blu-tack, so that its innermost in·ner·most adj. 1. Situated or occurring farthest within: the innermost chamber. 2. Most intimate: one's innermost feelings. n. corner is about at the centre of the cube. Carefully build a second cube as before (suspended from the new square); this cube is interlocked with the first cube but not connected to it, as in Figure 6. [FIGURES 5-6 OMITTED] Before we go any further, note that what we are doing is the three/four-dimensional analogue (electronics) analogue - (US: "analog") A description of a continuously variable signal or a circuit or device designed to handle such signals. The opposite is "discrete" or "digital". of what most people do when drawing a cube in two dimensions by first drawing two interlocking interlocking /in·ter·lock·ing/ (-lok´ing) closely joined, as by hooks or dovetails; locking into one another. interlocking Obstetrics A rare complication of vaginal delivery of twins; the 1st squares. Also observe that, at the moment, everywhere that edges meet they do so at right angles because in three-dimensional space we have all three right angles available and so far we have only built cubes cubes See QQQ. . Our next step is the three/four-dimensional analogue of drawing sloping lines to connect the corners of the squares in the two/three-dimensional cube. Connect a straw from the corner of one cube to the corresponding corner of the second, for each of the eight corners. In the process of threading through new and already positioned straws you will be able to tighten any imperfect imperfect: see tense. connections. Finally, adjust the length and location of the hangers hangers used for hanging x-ray films to dry. There is a clip type, with a clip at each corner, and a channel type in which the film sits in channels in the sides of the frame. (the lengths of cotton attached to the ceiling from which the tesseract is suspended) so that both of the original cubes are nice and square, and the diagonal struts A framework for writing Web-based applications in Java that supports the Model-View-Controller (MVC) architecture. Struts is deployed as JSP pages using special tags from the Struts tag library, which includes routines for building forms, HTML rendering, storing and retrieving data and are as parallel to each other as possible. This allows us to visualise what I tried to describe earlier: take a cube and move it along a sloping line that represents the extra right angle. The final eight straws that we have just placed allow us to fake the fourth right angle that we need for four-dimensions, in much the same way as we faked the third right angle when drawing a cube on two-dimensional paper. Where four lines meet at the vertices, we cannot have four mutually perpendicular lines in our three-dimensional world, and so some of them have to cheat by sloping off at an angle other than 90[degrees]. You should now have the finished model of a tesseract as in Figure 7 (see also Figure 8, which is a photo of the one that hangs outside my office door). Of course, what we have built is not really a tesseract, as they only exist in four-dimensional space, but it is a three-dimensional representation of one. In fact, talking about Figure 7 is quite hard: it is actually a two-dimensional representation (a drawing on paper) of a three-dimensional representation (the straw construction hanging in space from the ceiling) of a four-dimensional object (the real tesseract in four-dimensional space with four mutually perpendicular directions apparent at the vertices). Pretty scary scar·y adj. scar·i·er, scar·i·est 1. Causing fright or alarm. 2. Easily scared; very timid. scar stuff! [FIGURES 7-8 OMITTED] If you build such a model--or if you have a vivid imagination and can work with the diagrams--you should be able to pick out eight cubes: the two original right-angled cubes, and an additional six cubes which have square and parallelogram parallelogram, closed plane figure bounded by four line segments, or sides, with opposite pairs of sides parallel and equal in length. The rhombus, rectangle, and square are special types of parallelograms. faces (and which therefore appear as parallelopipeds). The two-dimensional diagrams and photo do not do the object justice, however. By looking at the three-dimensional model you will get more of a sense of the structure, especially if you move around it and take advantage of the fact that the model will flex, so you can temporarily make the parallelopipeds appear more cube-like. In fact, the three-dimensional model does not do justice to a real tesseract either. Just as a two-dimensional drawing of a cube has serious shortcomings A shortcoming is a character flaw. Shortcomings may also be:
Neighbours is a long-running Australian soap opera, which began its run in March 1985. ; similarly the three-dimensional model of the tesseract cannot easily show that each cube has six other cubes as its neighbours and that you can pass through the walls of one into the next. One final note, again limited by my capacity to show things only in two dimensions. There is an alternative way of drawing a cube that is shown in Figure 9. Instead of having two squares the same size we show one "nearer" (larger) and one "further away" (smaller), but then join corresponding corners as before. (For those of you who know your graph theory graph theory Mathematical theory of networks. A graph consists of vertices (also called points or nodes) and edges (lines) connecting certain pairs of vertices. An edge that connects a node to itself is called a loop. this is a planar graph In graph theory, a planar graph is a graph that can be drawn so that no edges intersect (or that can be embedded) in the plane. A nonplanar graph cannot be drawn in the plane without edge intersections. representation of a cube.) [FIGURE 9 OMITTED] We could use a similar approach to building a model of a tesseract. Take two cubes, one inside the other, and then connect corresponding vertices. A colour-coded depiction of this is shown in Figure 10; this is an alternative two/three-dimensional representation of a tesseract. You might like to think about whether this representation provides us with any additional understanding of the fourth dimension. [FIGURE 10 OMITTED] I hope this has helped you to visualise the fourth dimension, at least to some extent. Can you imagine what the fifth dimension is like or explain how you might show a five-dimensional hypercube? It is amazing a·maze v. a·mazed, a·maz·ing, a·maz·es v.tr. 1. To affect with great wonder; astonish. See Synonyms at surprise. 2. Obsolete To bewilder; perplex. v.intr. stuff. If you are interested in learning more, try some of the following. Abbot, Edwin A. (1952). Flatland flat·land n. 1. Land that varies little in elevation. 2. flatlands A geographic area composed chiefly of land that varies little in elevation. : A Romance of Many Dimensions. (6th rev. ed rev. abbr. 1. revenue 2. reverse 3. reversed 4. review 5. revision 6. revolution rev. 1. revise(d) 2. .). New York New York, state, United States New York, Middle Atlantic state of the United States. It is bordered by Vermont, Massachusetts, Connecticut, and the Atlantic Ocean (E), New Jersey and Pennsylvania (S), Lakes Erie and Ontario and the Canadian province of : Dover. (This is a story about a two-dimensional being's experiences in a three-dimensional world, and helps us appreciate the difficulty of comprehending four-dimensions.) Heinlein, Robert A Heinlein, Robert A(nson) (born July 7, 1907, Butler, Mo., U.S.—died May 8, 1988, Carmel, Calif.) U.S. science-fiction writer. He pursued graduate study in physics and mathematics and began his writing career in the pulp magazine Astounding Science Fiction in the 1930s. . And He Built a Crooked House
Stewart, Ian (2001). Flatterland: Like Flatland, Only More So. London: Macmillan. (Presents a wide variety of different geometries.) Author's note Parts of this article first appeared in Delta, the journal of the Mathematical Association The Mathematical Association is a professional society concerned with mathematics education in the UK. It was founded in 1871 as the Association for the Improvement of Geometrical Teaching and renamed to the Mathematical Association in 1897. of Tasmania, Vol. 29 No. 1 (pp. 10-12), 1989. Helen Chick chick abbreviation for chicken (1). University of Melbourne
In 2006, Times Higher Education Supplement ranked the University of Melbourne 22nd in the world. Because of the drop in ranking, University of Melbourne is currently behind four Asian universities - Beijing University, <h.chick@unimelb.edu.au> |
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