Zometool Project Series: the world’s most powerful Green lines Details of tetrahedron (and fun!) modeling system. Kids, educators, and START HERE! Brace yourself corner (A) and conceived by octahedron corner Nobel-prize winning scientists all love Zometool: Clark Richert Cut across a square lawn and you’ll use a green (B) show green line symmetry in relation • it’s unique, brilliant, beautiful and designed line: the fastest way from one corner to the opposite to the cube. • all kits are compatible— more parts, more power! by Jean Baudoin corner of a square. Nail a cross-brace to a square frame — you’ll also use a green line: the fastest way • guaranteed for life! & Fabien Vienne from an unstable shape to a rigid structure. “The mind, once stretched by a new idea, never regains its origi- Why not make balls with 30 nal dimensions.” – Oliver Wendell Holmes Take 6 braced squares, extra holes to fit all the greens? join the edges and you Well, they’re hell to make, and balls with so many holes result Expand your Universe! This kit get a cube with a tetra- hedron inside (see in “ball-heavy” models. Most * adds new dimensions to your Zometool Tetrahedron, inside.) people agree that green lines collection and new horizons to your mind. Find 5 such cubes in a work pretty well, once you get 1 dodecahedron, and you to know them . When combined with other Zometool kits, begin to see the power you can build: of Zometool Green Lines! Five tetrahedra associated with Completing Plato (See 5 Tetras, inside.) 5 cubes of the dodecahedron. • Regular tetrahedra and octahedra This kit is designed to supplement the blue, yellow and red lines in Zometool System kits. You can • Archimedian solids Not easy being green • Relationships among Platonic Solids build the first two models shown on the other side using only parts included in this kit. The octahedron • Amazing duals, compounds etc. Look at the end of a green strut. It has a diamond cross-section, yet goes into a pentagonal hole, the and the tetrahedron are two of 5 regular (Platonic) • Oct-tet trusses and much more! same hole a red strut uses. Because it’s “kinked,” solids built with green lines (you can build the other Warning! This ADVANCED KIT is for each green line can take 5 different positions in one three—the cube, dodecahdron and — using blue lines.) experienced Zometool users only! pentagonal hole. Try it for yourself: *30 new dimensions, actually The rest of the models on the other side are cited in Zome Geometry, the Zometool textbook. Title bars US Patents RE 33,785; MADE IN USA for these models are yellow, and the numbers in from kid-safe materials 6,840,699 B2. Zometool is a registered each title bar refer to the rel- trademark of Zometool evant section of the book. You Inc. Based on the don’t need to buy the book in 31-zone system, 5 positions of a green line discovered by Steve order to build the models— Baer, Zomeworks You can use a framework built from blue lines to you just need enough Zome- Corp., USA © 2009 orient yourself. Note the position of the green lines tool components in your in relation to blue lines to “stay on track” with more collection (there are enought WARNING: complicated models. greens in this kit to build all the O-3 ! Choking Hazard models inside). SMALL PARTS. NOT for children under 3 years. 888-966-3386 1

When greens get the blues cube, truncated tetrahedron and truncated octahe- While the length of a green line (√2) seems a new periodic table Jean Baudoin, a French dron shown on the other side are examples of these to be the “first” irrational number, a wor- of the elements, by artist who independently You can’t build a regular octagon (the STOP sign almost semi-regulars. thy challenger is the Divine Proportion packing truncated discovered elements of shape) with this kit. You would need some struts ((√5+1)/2), also known to the Pythagoreans. octahedra in a body- Zometool geometry in with the same length as a blue line, but the same di- Irrational Greeks With Zometool, you get both: green lines centered cubic lattice the 1960s, used green rection as a green line. We call them “blue-greens,” come in Divine Proportion powers! (I.e., a (BCC), a common directions in many of his Models Chris Kling (informed by Steve Rogers), Brian Hall, but we hardly make them, because people hardly How long is a green line? short green plus a medium green are the structure in crystals sculptures. He became a strong advocate of and Scott Vorthmann give a taste of green lines’ versatility. use them .2 Ancient Babylonians provided same length as a long green, and the ratio (see Truncated Octa- adding green lines to Zometool (along with an approximation to 4 decimal of a short to a hedron, inside.) Baer other implied directions6 ) and joined forces Endnotes In addition to the places, but the Greek philoso- 1 For models you can’t build with Zometool, try TubeSpace, designed by medium is the supplied struts cut to with designer Fabien Vienne to draft the cur- Fabien Vienne and Jim Hausman (see http://www.emergentworld.com) regular octagon, pher Hippasus discovered that same as a me- the proper lengths but rent green lines. Zome nuts celebrated the 2 Order blue greens at 888-966-3386. But don’t say we didn’t warn you! you would need whatever the accuracy, it was 3 “A square plot of ground 10’ long by 10’ wide contains a 100 square feet; dium to a long.) they pirated holes from first parts with a Green Party at Baudoin’s ...to double this... we must find the length of a side of [a] square, so that its blue-greens to never exact (“by numbers this the red lines (a tradition Moulin de l’Image, near Chartres, in May 1999. area... is two hundred feet. By numbers this cannot be done; for if the sides make all five regular 3 are 14 feet, these give 196 square feet; if 15 feet, they give a product of 225. cannot be done”. ) A disciple of Pythagoras, he Green history we still honor!). Richert Marcus Vitruvius Pollio, de Architectura, Book 9. (Platonic) solids with the same edge length, and found that the square root of 2— the length of named the show “The 4 Richert and associates conceived of “the first hippie commune,” , in Spring of 1964. “We envisioned a whole city as a live-in work of 4 Forever Green some of the semi-regular (Archimedean) solids. But a green line—cannot be expressed as a ratio of Artist Clark Richert 61-Zone Truss”, since the green lines add 30 Drop Art.” Built on 7 acres of pasture near Trinidad, Colorado, Drop City was none of these shapes are part of Zometool geom- whole numbers. added green lines to Green lines may be used to build virtually an awarded R. Buckminster Fuller’s Dymaxion Award in 1969. new directions to Zometool’s 31 blue, yellow 5 The physical embodiment of the 31-zone system, Zometoy was developed 7 etry. So even though you could use blue greens to Zometool geometry. and red directions. infinite number of models. ZomeSpace will by Steve Baer and associates of Zomeworks Corporation in the early 1970s. The Pythagoreans had preached that all numbers Zometoy kits consisted of soft polyethylene “Whiffle” balls and wooden build a stop sign, they’re really a dead end! With Around 1976, he offer a rich environment to explore and share “Tinkertoy” struts in a wooden case, with Baer’s Zome Primer as a manual. blue greens, the parts don’t line up the way they do could be expressed as ratios. Though Hippasus was invited (through green line (and other Zometool) models with 6 Baudoin and Vienne called these other lines the “bleu, jaune, et rouge was right, the Pythagoreans treated his discovery as symmetrique” (blue, yellow and red symmetricals). They are represented in real Zometool structures. Steve Baer) by the enthusiasts, teachers and professionals in Scott Vorthmann’s vZome program as black, purple and orange lines, You can substitute green lines any place you would religious heresy and either exiled or murdered him. and in conjunction with the 61-zone system comprise the 181-zone system. Art Research Center around the world. Still other lines have been identified by Vorthmann and associates. Legend has it that while at sea his fellow Pythagore- use blue-greens and build pretty good approxima- to create a Zometoy exhibit in Kansas City. 7 ZomeSpace model database is in development as of this writing (3/2009). See inside for model-building instructions, tions of the semi-regular solids, etc. The truncated ans threw him overboard. Richert’s packings of cubes (a), rhombic dodecahedra Used with ZomePad software, it allows you to share models and ideas with Richert used the opportunity to envision (b) and truncated octahedra (c). Build them all (inside)! shadow activities, and more! the global circle of Zometool users. See www.zometool.com for more info. 2 3 Octahedron Tetrahedron Shadows Tetrahedron

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Also step 1 for 5 Tetras

ZOMETOOL RULES: 1. Place a blue or yellow 2. Don’t break it 3. Don’t crush models. Zometool Green Line Project — Clark Richert, concept; Fabien 5 Tetras 11.3 apart; take it apart! strut* in any hole. Vienne and Jean Baudoin, green line design; Dr. Scott Vorthmann, 1. If it works, it works perfectly. 2. Point the strut at the sun. Our mission: vZome software for images; Anni Wildung, graphic design; 3. Cast shadow on a board • make learning fun Paul Hildebrandt, copywriting and project management. Contact at 90º to sun’s rays. • create value [email protected]. Based on the 31-zone system discovered *Use a red strut to get a 5-fold by Steve Baer, Zomeworks Corp., USA. © 2009 Zometool Inc. shadow (see 5 Tetras). • build a better world

Truncated Cube* 3.4* Truncated Tetrahedron 3.4 2 3 4

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Hexagonal Slice Through Cube 3.1.5 Truncated Octahedron 3.4

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3.2.3 12B *Models in this section are from Zome Geometry; numbers in Stella Octangula Cuboctahedron & Rhombic Dodecahedron the yellow title bars refer to Explorations or Challenges in the text. All models can be built using this Green Lines kit and any 1 2 3 1 3 5 Zometool System kit with sufficient parts. Zome Geometry is the only textbook especially for Zometool! It’s written for classroom use with the Advanced Math Creator 4 kit, equivalent to two Creator 2 kits plus this Green Lines kit. Available at www.zometool.com.

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