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Introduction to Zome a New Language for Understanding the Structure of Space

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Introduction to Zome a New Language for Understanding the Structure of Space 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 Zome’s 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. There’s 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 it’s built with number 2s (blue struts) in Zome, the number 3 keeps cropping up.1 1 I believe it’s 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 come 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 it’s 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 – (a) (b) another way of showing the length of the yellow strut is intimately related to the number 3 (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? Let’s 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. It’s 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, it’s 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 color’s 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. That’s one step toward understanding Zome: it’s 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, you’d probably use the same algorithm: put a point in the middle and connect the dots (b). But you’d be wrong€46 (a) (b) Bubbles always shows the most efficient solution (a). It’s 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 5 seeing which one is shorter. 4 The 3-dimensional bubble 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 (a) (b) (c) space is the rhombic dodecahedron (b), which fills space like a 3 dimensional hexagon (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 (a) (b) thought of as a shadow of a 4-dimensional number 3, as discussed below. The carbon structure of diamond can be seen in a bubble made with an octahedral frame,7 showing five tetrahedra joined in a manner that’s 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 (a) (b) systems to super-efficient batteries for hybrid cars. 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, it’s 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-dimensional triangle, or a simplex.
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