Why are Green Lines so useful?
Green Lines "complete" the Zometool... so you can build all 5 Platonic Solids (and their intimate relationships in space)! Build the tetrahedron (4-faces) and octahedron (8-faces) with Green Lines; the cube, dodecahedron and icosahedron with Blue Lines. Or build them all together, like Conway did.
John Conway (to the right of Kepler) discovered a most intimate nesting of the 5 Platonics which he called his Cosmogram (and we call Kepler's Obsession or Kepler's Kosmos). There are lots of ways these "perfect" solids like to make love -- see the love among the cube, tetrahedron and octahedron above. And it sure helps to use the cube(s) for scaffolding framework when you're learning to use Green Lines.
Why are Green Lines so hard to use?
Green Lines take 5 positions in any "red' hole ...so it's easy to choose the wrong direction. The fastest way to orient a Green Line is to use Blue "scaffolding." You can stabilize a square made from medium Blue (B1) struts using a medium Green (G1) strut. Position the strut between balls on 2 opposite corners so that the stubs line up with the pentagonal (red) holes. You can stretch the square to get the strut in position, then push the ends in. If you turn your square into a cube and "connect the dots" with Green Lines, you get a tetrahedron inside.
Start in one cube, using 6 of the 30 Green Lines. The six lines are part of the 13-zone system,* that includes the 3 Blue Lines, 4 Yellow Lines and 6 Green Lines. A Blue cube or Blue scaffolding can help you find the the right green lines for building most models with 2- and 3-fold symmetry. Green Lines live in all 5 cubes of the dodecahedron (6 Greens x 5 cubes = 30 Green Lines). The model of the 5 tetrahedra (last frame) contains all 30 Green Lines
*see Peter Jon Pearce (Synestructics), who informed the design of the Zometool
How can Green Lines be in the Silver and Golden Proportions?
You can find the Silver rectangle by "cutting" a cube along a diagonal (Green) plane. If you cut a Silver rectangle in half, you get another Silver rectangle (√2:1=1:√2/2). European paper mills applied the Silver Proportion to their sheet sizes, with practical advantages of reducing waste cut-off, and that reducing content on a larger sheet fits perfectly on a smaller sheet. Because of the dual relationship among Greens and Blues, we make Half Green (HG) struts (in the center of the 4th image above).
Green lengths are in the Golden Proportion: if a Blue Line has length 1, then a Green Line has a length of √2 (or √2/2: Half Greens). Like other lines in the Zometool, relative lengths of the Green Lines are in the Golden Proportion (i.e., a short Green plus a medium Green are the same length as a long Green, and the ratio of short to medium is the same as medium to long). You can make a Golden Rectangle with Green Lines (just like Blues, only √2 bigger!) Conway's Cosmogram (Kepler's Obsession) features on octahedron with its edges intersected in the Golden Proportion by an icosahedron.
Why can't I build a STOP sign with Green Lines?
We don't make Blue Greens anymore... so you can't build a regular octagon with the Zometool. You would need struts with the length of a Blue Line and the direction of a Green Line (or vice versa -- Blues and Greens are duals!). We used to make Blue Green Lines, so people could build the regular octagon and some of the Archimedean solids. But Blue Greens are not part of Zome geometry, so even though you could use them to build a STOP sign, they are really a dead end. Besides, you can make Archimedeans (like the truncated cube, above) that are damn close, using the right combinations of Blues, Greens and Half Greens.
Were Green Lines the "cause" of one man's demise?
Greek Pythagorean philosopher Hippasus discovered that the length of a green line is an irrational number (√2= 1.414... ). Ancient Babylonians provided an approximation to 4 decimal places, but whatever the accuracy, it was never exact ("by numbers this cannot be done", i.e., the square root of 2 cannot be expressed as a ratio of whole numbers). This was heresy for the Pythagoreans, who preached that all numbers could be expressed as ratios of whole numbers. Legend has it that while at sea, they threw Hippasus overboard and he drowned.
Who "discovered" Green Lines?
Artist Clark Richert added Green Lines to Zometool geometry around 1976. Invited by Steve Baer to do a Zometoy exhibit at the Art Research Center in Kansas City, Richert used the opportunity to envision a new periodic table of the elements, by packing truncated octahedra in a body-centered cubic lattice. Baer cut struts to the proper length, but they pirated holes from the red lines (just like us!) Richert called the show The 61-Zone Truss, since Green Lines add 30 directions to the Zometool's 31 Blue, Yellow and Red Lines). Fabien Vienne and Jean Baudoin brought Green Lines to fruition, and we celebrated with a Green Party near Chartres in 1997.
What's the difference between Naked and Clothed Green Lines?
The only difference is the jar (and lid). Your Naked Green Lines kit comes in a 2 mil poly zip bag instead of a jar (and lid). You get the same parts, same instructions and same good karma for less money when you buy Naked Green Lines.
Steve Rogers recently built the model in frames 1-7 above, the Truncated Tetra w/Greens. It lives in one of the 5 cubes of the dodecahedron. If you were to build all 5 TTw/Gs, you'd get the model in the last 3 frames (shown in 2-, 3- and 5-fold "shadows" -- easy to do in vZome). To make this exact model, you would need parts we don't make yet (double cross blue and green "Bobs," and supershorts) but it can be built larger with balls at all the crossings. Time to buy more parts!