Introduction to Zome A new language for understanding the structure of - PDF document

Introduction to Zome A new language for
Introduction to Zome A new language for understanding the structure of space Paul Hildebrandt Abstract This paper addresses some basic mathematical principles underlying Zome and the importance of numeracy in education. Zome geometry Zome is a new language for understanding the structure of space. In other words, Zome is hands-on math. Math has been called the queen of the sciences, and Zomes simple elegance applies to every discipline represented at the Form conference. Zome balls and struts represent points and lines, and are designed to model the relationship between the numbers 2, 3, and 5 in space in a simple, intuitive way. Theres a relationship between the shape of a strut, its vector in space, its length and the number it represents. Shape Each Zome strut represents a number. Notice that the blue strut has a cross-section of a golden rectangle -- the long sides are in the Divine Proportion to the short sides, i.e. if the length of the short side is 1 the long side is approximately 1.618. The rectangle has 2 short sides and 2 long sides, has 2-fold symmetry in 2 directions. Everything about the rectangle seems to be related to the number 2. If the blue strut represents the number 2, it makes sense that you can only build a square out of blue struts in Zome: the square has 2x2 edges, 2x2 corners and 2-fold symmetry along 2x2 axes. It seems that everything about the square is related to the number 2. What about the cube? Vector -- It seems logical that you must also build a cube with blue struts, since a cube is made out of squares. But a cube has 2x3 faces, 23 corners and 2x2x3 edges. It also has 3+2x2+2x3=13 symmetry axes. So although its built with number 2s (blue struts) in Zome, the number 3 keeps cropping up.1 1 I believe its difficult for most people to see the number 3 in the cube because of the way we encounter the cube as children, i.e., as the basic unit of kindergarten blocks. Blocks are normally oriented to gravity; if we could easily balance them on one corner, the 3-fold nature of cubes would be obvious. Which means you should be able to find a yellow (3-fold) line in the cube. The natural place is the corner, where 3 squares c

ome together. Notice that you can rotate
ome together. Notice that you can rotate the cube three times on the yellow strut and then it returns to its original position. In other words, the yellow line is a 3-fold rotational symmetry axis of the cube. Another way to see it is to cast a shadow. If you point the number 3 (yellow) line directly at the light source and cast a shadow on a surface that is perpendicular to the rays of light, you get a hexagon. In other words, you get a 2-dimensional version of the number 6, or 2x3 in 2 dimensions. It has 6 corners, 6 edges, six spokes going to its center and its formed from 6 equilateral triangles. Length --This indicates a relationship between the shape of the strut, its vector in space and the number it represents. But what about the length of the strut? You can follow the 3-fold symmetry line from one cube corner to the opposite cube corner. So the length must be (by Pythagoras) the square root of 12 + 12 + 12, or root 3, divided by 2, which is cosine 30º (a).2 You can see that a yellow strut is exactly the height of an equilateral triangle built with blue struts in Zome another way of showing the length of the yellow strut is intimately related to the number 3 (b). (a) (b) But what about the red struts? These represent the number 5 in Zome. We saw that the cube is made from the numbers 2 and 3, but can we find the number 5 in the cube? Lets try casting a shadow along the number 5 line, like we did with the number 3 line. No matter where you insert the red strut, you always get the same shadow. Its an interesting rectangle: the long edge measures 1/(cosine 18º) or 1/cosine (PI/2x5). Another way to find the number 5 in the cube is to build a roof on it. I can use shorter blue struts to put a special hip roof on one of the cube faces. If I put the same roof on all of the faces of the cube I get a dodecahedron, a beautiful polyhedron made of 12 regular pentagons. Now we can see that a red strut pierces the center and is perpendicular to each pentagonal face.3 2 The relative lengths of Zome struts are unity (blue), cosine 30º (yellow), cosine 18º (red) and cosine 45º (green). So although I call the blue strut a 1-dimensional number 2, i

ts not because its length is 2, but beca
ts not because its length is 2, but because it represents a 2-fold symmetry axis in space. 3 The pentagons lie in red planes. Red planes are always perpendicular to red lines and exhibit 5-fold symmetry in Zome. Likewise blue planes are perpendicular to blue lines and yellow planes are perpendicular to yellow lines, and each exhibits that colors corresponding symmetry. If you cast a shadow of the dodecahedron along the red line, the outside shape becomes a decagon. It has 2x5 (i.e., blue x red) edges and 2x5 vertices (a). In the second image you can still see the odd projection of the cube with the 1 by 1/ cos18º proportions inside the dodecahedron (b). (a) (b) The length of the red strut can be seen as the height of a regular pentagon; in other words, cosine 18º, or the cosine of Pi/2x5. So we see that the cross-section, vector and length of the red strut are all intimately related to the number 5, just as the yellow strut is the number 3 and the blue strut is the number 2. Thats one step toward understanding Zome: its the relationship between the numbers 2, 3 and 5 in 2-, 3- and higher dimensions. Bubbles You can illustrate some interesting minimal surface problems using Zome and soap solution. For example, what if you needed to connect three cities using the minimum amount of cable? You might start with a triangle (a), but of course you can remove one of the sides and the 3 points are still connected (b). (a) (b) Bubbles show a more efficient solution: each cable is goes to the center of the triangle (a).4 Asked to solve the problem for four cities corresponding to the corners of a square, youd probably use the same algorithm: put a point in the middle and connect the dots (b). But youd be wrong€46 (a) (b) Bubbles always shows the most efficient solution (a). Its interesting that this exact graph (b) is contained in the projection of the cube-dodecahedron model along its 2-fold (or blue) axis of symmetry (c). (a) (b) (c) You can verify this mathematically or test it in a hands-on method by laying the corresponding Zome struts for each solution end to end and seeing which one is shorter.5 4 The 3-dimensional bubble

made with a triangular prism is flatten
made with a triangular prism is flattened on to a 2-dimensional surface to find the solution, shown here by projection onto a screen along a blue (2-fold) axis. Bubbles can also be used to illustrate closest packing problems, such as the one so elegantly worked out by bees building a beehive (a0. The ideal shape for fitting the maximum number of bees in a minimum of space is the rhombic dodecahedron (b), which fills space like a 3 dimensional hexagon (c). (a) (b) (c) This shape can also be thought of as the outer shell of a 4-dimensional cube shadow.6 The packing of rhombic dodecahedra is closely related to the diamond structure. You can illustrate a carbon atom, with a bubble in the shape of a tetrahedron (a), or a 3-dimensional number 3. The whole model (b) can be thought of as a shadow of a 4-dimensional number 3, as discussed below. (a) (b) The carbon structure of diamond can be seen in a bubble made with an octahedral frame,7 showing five tetrahedra joined in a manner thats topologically equivalent to diamond structure found in nature. The unit cell of the diamond structure is shown in (b) (a) (b) Zome can also model graphite, and any number of new carbon structures theorized by Japanese scientists in the 50s and confirmed by Americans in the 80s, such as buckyballs (a) and nanotubes (b). Nanotechnology promises a whole chemical candy store of new applications from better drug delivery systems to super-efficient batteries for hybrid cars. (a) (b) 5 You can build the incorrect intuitive solution with Zome half green struts (of length cosine 45º.) Assuming a square edge length of 1, the first solution uses 4 x cos. 45º= 2.828 units of cable while the bubble solution uses 4 x cos. 30º + -1 = 2.759 units 6 Shadow means projection here. 7 The octahedral frame will form several different bubbles. Although the diamond structure seems to be the most natural, or stable, form, its a bit more difficult to find (perhaps mimicking the occurrence of diamonds in nature!) You can illustrate 4-dimensional figures with bubbles: as mentioned, the

tetrahedron bubble is a shadow of a 4-d
tetrahedron bubble is a shadow of a 4-dimensional triangle, or a simplex. In Zome, we think of the yellow line as a 1-dimensional number 3, while a triangle is a 2-dimensional 3 and a tetrahedron is a 3-dimensional 3. So, 2 points define the 1-D #3, 3 lines form the 2-D #3, 4 triangles form a 3-D #3, and 5 tetrahedra form a 4-D #3.8 A cube bubble yields a shadow of a 4-dimensional square, or a hypercube. In this case, 2 points define the 1-D #4 (a line), 4 lines form the 2-D #4 (a square), 6 squares form a 3-D #4 (a cube), and 8 cubes form a 4-D #4 (a hypercube). Just as a 3-dimensional cube will cast any number of 2-D shadows, this shadow of the 4D cube is just as valid as the beehive example mentioned above.9 And the dodecahedron bubble is the seed of the 4-dimensional pentagon, also known as the hyperdodecahedron, or 120-cell. vZome demonstration It takes about two hours to build the 120-cell shadow using Zome parts, but you can do it in a few minutes with vZome10, the virtual Zome program being developed by Scott Vorthmann. Just as the dodecahedron is formed from 12 pentagons joined edge to edge, the 120-cell is formed by 120 dodecahedral cells joined face to face. (a) (b) If you are familiar with the model, the easiest way to build it is step-by-step, using symmetry operations. This projection of the 120-cell has icosahedral symmetry, or a combination of the 2, 3, and 5-fold symmetries above. If you perform this operation on a red (5-fold) strut, you get a total of 12 red struts (5x12=60.) If you perform it on a yellow (3-fold) strut, you get a total of 20 (20x3=60), and the operation will yield 30 (2-fold) blue struts (2x30=60). The icosahedral symmetry operation will turn one green strut into 60, leading one to believe that the green strut represents the number 1 in Zome space(1x60=60.) (c) (d) 8 This series is mathematically consistent, and you could conceivably build Zome models of these tetradehral polytopes in up to 61 dimensions. 9 The bubble example is a cell-first perspective projection of the hypercube while the body-centered rhombic dodecahedron is a point-first parallel projection. 10 Currently, vZome can be downloaded fr

ee at www.vorthmann.org in Java Webstar
ee at www.vorthmann.org in Java Webstart format. (a) (b) The Zome shadow of the 120-cell is called a cell-first projection. In practical terms, this means the center of the model will be a regular dodecahedron. In fact, this is the only regular cell in the model. All the others are distorted by projection. I call them the fat red cell (which is squashed along the red (5-fold) axis), the yellow cell (which is squashed along the yellow (3-fold) axis), the thin red cell (also squashed along the red (5-fold) axis) and the flat blue cell (which is the 2D shadow of the regular dodecahedron that weve seen several times.) In this model, only 75 of the cells are visible; the rest of them line up exactly with the ones you can see. (c) (d) (a) (b) To understand how the cells become squashed by their projection from 4 to 3 dimensions, its useful to think of an analogous case from 3 to 2 dimensions. Say a pentagon is a cell of the dodecahedron. If we view the pentagon along its red (5-fold) symmetry axis, we see a regular pentagon (a). But if we rotate the pentagon it becomes more and more squashed(b) & (c), until, when its perpendicular to our viewing plane, it becomes a line segment (d). Its been squashed from 2 to 1 dimensions. The Zome shadow of the 120-cell is built from only these 4 pentagons. (c) (d) Likewise, the regular dodecahedron in the center of Zome 120-cell model is in a hyperplane11 that is parallel to our 3-dimensional world. As we move out toward the surface of the model, the hyperplanes associated with each type of cell are at successively greater hyperangles to our world, until we reach the flat blue cells, which are in a hyperplane perpendicular to our 3-dimensional space. To build this model in vZome, well start by building the regular dodecahedron in the center. Three pentagons join at each vertex of the dodecahedron, so we follow two yellow (3-fold) lines from the center of the model to two adjacent vertices of the cell. Each yellow line is the equivalent to the cube diagonal mentioned above (3 squares join at every vertex of the cube.12) By joining the vertices we get one edge of the first dodecahedral cell in the 120-cell. 11

Hyperplane is another term for a 3-dime
Hyperplane is another term for a 3-dimensional space. 12 Every dodecahedral vertex actually corresponds to 3 cube vertices, since a dodecahedron defines 5 cubes. You work it out! (a) (b1) (b2) (c1) (c2) (d1) (d2) (e) By finding one edge of the dodecahedron, by symmetry, weve found all of them (a). Following this method, we can build the 120-cell, strut by strut. The regular cell is surrounded by 12 fat red cells (b). The fat red cells are surrounded by 20 yellow cells (c). These are surrounded by 12 thin red cells (d). And these cells also form the flat blue cells on the surface of the model (e). Notice that these are the same as the shadow of the dodecahedron cast along the blue (2-fold) axis, which weve seen before. Its certainly a beautiful model, but what relevance does it have to our world? The ancient Greeks called the dodecahedron the 5th element, and reserved it to represent the shape of the heavens. Johannes Kepler believed the nested 5 Platonic solids determined the distances between the known planets, and in proving himself wrong established basic laws of modern cosmology. And a recent interpretation of radio telescope data suggests that our universe may be finite and shaped like a house of mirrors resembling the hyperdodecahedron!13 Using the vZome program you can build a 120-cell out of 1s and 0s in a few minutes.14 Numeracy and education If we can build a virtual world out of 1s and 0s, think what we can do with the numbers 2,3, and 5! They are the first 3 prime numbers and the first 3 non-digital numbers in the Fibonacci sequence, which naturally leads to the Divine Proportion the governing relationship in Zome In fact, these numbers can be thought of as the building blocks of our natural and human-built world. In addition to examples such as nanotechnology and the theoretical shape of the universe, we find these relationships in natural growth, biotechnology, the Coopernican revolution in material science launched by the discovery of quasicrystals, the subatomic structure of the atom, artificial intelligence, new engineering and architectural structures, to name a few. As our understanding of this language grows, it can lead us to knowledge and wealth beyond our present

comprehension.
comprehension. 13 Weeks, et al, Nature, 9 October, 2003 14 My father (Dr. T.W. Hildebrandt), who worked on the first electronic digital computer during the Princeton IAS project, reports that Von Neumann foresaw image processing as one use of computers in the distant future. Ive called Zome a new language for understanding the structure of space, but you could also call this type of understanding numeracy. And I believe numeracy is going to be as important in the 21st century as literacy has been in the preceding one. Literacy -- In order to learn how to read, we had to jump through two levels of abstraction (from the thing to the sound representing it, and then from the sound to abstract symbols representing the sound) and then learn a complicated set of rules, full of exceptions, for putting the symbols together. All this before most of us could pour a glass of milk for ourselves. No wonder until recently, reading was a highly prized talent reserved for the privileged elite. Now, its become a survival skill. Literacy one of the most important factors in raising the standard of living for people everywhere, and certainly the determining factor in childrens ability to get ahead in life. Everyone naturally tends toward literacy, because we love stories. We start telling our children stories before they can speak, and patiently nurture the love of storytelling for years before slowly introducing the nuts and bolts of reading. Its a huge undertaking. But when we succeed in building a bridge between kids natural love of stories and the technical challenges of literacy, magic happens that enriches children both materially and spiritually for their entire lives. Numeracy -- In the same way, everyone has natural tendency towards numeracy. We love patterns for example, you can feel the number 2 when you drive over railroad tracks: one-two, one-two. Or in the poetry of childrens books like One Fish Two Fish, Red Fish Blue Fish. In fact, most of us encountered the number 2 soon after we entered the world, in the 2-fold reflection symmetry of our mothers faces. Simple things like rhythm and balance are the basis of the way we feel numbers without even thinking about

them. Thats number sense, our natural
them. Thats number sense, our natural love for numeracy. But who among us takes the time to build a bridge between the kids natural love of numbers and the abstract math assaulting them on the first day of school? Mathematicians, engineers and scientists value numeracy more than most, and may encourage their own children to build on their intuitive number sense. Some other children who are especially gifted may work on their own to build a connection between their innate number sense and the abstract symbols scratched on the chalkboard. The rest are likely to accept math as necessary drudgery, or worse, develop math anxiety. In my culture, numeracy is hardly valued at all. Most Americans would return a blank stare if you asked them to define numeracy.15 As a result, math anxiety is passed on from 15 which should come as no surprise in a country where we spend more money on lawn care than on public education!) generation to generation. Children who were taught to fear and hate math grow up to become teachers who infect their students with a similar revulsion for math.16 Zome is a tool to bridge the gap. Children can feel the shape of the numbers 2, 3 and 5 in their fingertips when they handle the blue, yellow and red struts. As Zome cofounder Marc Pelletier says, theres an instruction manual in every ball. But the tool by itself is not enough to create a culture of numeracy any more than a piano builds an appreciation for music in a family where no one can play the instrument. We need to carefully cultivate numeracy in our children, public schools, and institutions if we want to reap the rich harvest it promises. The whole world is math -- streaming into our eyes and ears all the time. Einstein said, "Imagination is more important than knowledge." But when we join the two the imagination of our intuitive numeracy with knowledge contained in the severe beautyof abstract math we begin to tap the awesome power of the human spirit to build a better world. 16 Children who are especially gifted with an intuitive number sense, but never bridge the gap to abstract math, often grow up to be artists and musicians.