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Home , Harmonices Mundi, Katharina Kepler

Zometool Project Series: the world’s most powerful (and fun!) modeling system. Kids, educators, and ’s START HERE! World in turmoil assistant, mapmaker, astrologer, calendar- version. Kepler “got it” almost instantly Nobel-prize winning scientists all love Zometool: Obsession maker… even as a lawyer (to save his and defended Copernicus in a public de- • it’s unique, brilliant, beautiful model Imagine a world in which short-sighted mother from being burned as a witch!)1. bate, ensuring he would never get a job in • all kits are compatible— more parts, more power! designed by Dr. John rulers use religious intolerance to create He became Imperial Mathematician in Tübingen—Martin Luther himself rejected • guaranteed for life! Conway “endless war,” bankrupting their people. Prague, though he wasn’t always paid.2 Copernicus’s scheme, using the Bible to “The mind, once stretched by a new idea, never regains its original dimensions.” – Oliver Wendell Holmes Ancient practices like herbology, witch- “disprove” it. craft and continue to flourish was both profoundly re- amidst breathtaking advances in science ligious and a rigorous scientist. Like A deeply religious and rigorous scientist and Pythagorus, Kepler believed that God The flash of light of the Renaissance, JOHANNES KEPLER and technology. Legal scholars wrestle made the universe according to a plan, believed the relationships among the 5 with definitions of torture and its “proper” Kepler got a job 677 km (421 miles) away, and he felt it was his Christian duty to un- “perfect” 3-dimensional shapes (Platonic use, while new information technology is as professor in Graz (now in Solids) governed order in the universe. engulfing the masses in a tsunami of text derstand God’s works. His mathematical Austria). While teaching math he was brilliance and love of truth made his work taken by a drawing of concentric circles Build a beautiful and elegant model of and images. We’re on the brink of a cul- relevant to this day while raising the dan- those relationships, and discover: tural revolution that will forever transform inscribed in and circumscribing a triangle, our place in the universe… der of the religious authorities in his day. believing it might be the “key to the uni- • What inspired Kepler verse” (i.e., how God determined • How the 5 “perfect” 3D shapes fit together It’s the dawn of the the distances between the of the • Why Kepler’s “mistake” laid the  Me and my big mouth 17th century, and foundations of modern astronomy ). He tested his theory, but the After graduating from Maulbronn3 Kepler • And how (in a sense) he might have  Johannes Kepler ratios were wrong. is about the lay the won a scholarship to the University of been right all along! Kepler’s first foundation of modern Tübingen, and was expected to become attempt to MADE IN USA US Patents RE 33,785; 4 from kid-safe materials 6,840,699 B2. astrophysics. a Lutheran pastor . geometrically Zometool is a registered define the trademark of Zometool Mathematical sciences (arithmetic, geom- distances Inc. Based on the In a life full of turmoil between 31-zone system, etry, astronomy and ) were required planetary discovered by Steve and personal tragedy, Baer, Zomeworks courses. Kepler’s astronomy professor orbits (with Corp., USA © 2009 Kepler made significant contributions in color-coding publicly taught that the was the cen- fields diverse as astronomy, cosmology, informed ter of the universe (Ptolmy) but privately Zometool). WARNING: 888- optics, calculus, logarithms, and geom- O-3 ! believed in Copernicus’s sun-centered Choking Hazard 966-3386 etry. He also worked as an innkeeper’s SMALL PARTS. NOT for children under 3 years. 1

Then he considered the problem in 3 thousand pages of hand-written calcula- Tycho (not a toy) geometry: a proof that there are only 13 Kepler Blocks Schectman’s dimensions. The circles of the planets’ tions. But it ultimately yielded his 3 Laws Archimedean solids8, 2 new non-convex discovery incited 5 orbits became spheres, and the shapes Rich, fat and noseless, Tycho Brahe kept of Planetary motion, which form the basis regular polyhedra, and the first orderly Kepler also discovered the rhombic a “Copernican separating them became the 5 regular sol- the most meticulous astronomical data on of modern astrophysics. treatment of mathematical tilings. triacontahedron and its sub-units (Kepler revolution”11 in ids. It was an elegant solution: there were the . When Kepler was kicked out of blocks), which are 3-dimensional cousins the materials sci- 6 Kepler’s Laws 2 non-convex only 6 known planets and the Greeks had Graz for being a Lutheran, Tycho invited regular polyhe- of Richert tilings. While Richert’s tiles are ences. We hope proved there are him to be his assistant in Prague. Tycho 1. Planets move in elliptical orbits with dra discovered Quasicrystal photo by Dr. H.U. 2 types of “squashed” squares (paralle- J.K. is chuckling Nissen. The “apple blossoms” only 5 regular jealously guarded his own data, which he the sun at one focus. by Kepler and published in grams), the triacontahedron can be built in heaven. approximate Kepler blocks. 2. In their around the sun, planets polyhedra. The hoped would prove that the sun rotates Harmonices up from 2 types of “squashed” cubes sweep out equal areas in equal times. mundi in 1619. model’s beauty around the earth while all the other planets (parallelepipeds), and Kepler blocks will 3. The squares of the times to complete Endnotes and simplicity go around the sun. He set Kepler to work is a brilliant and daring 1 one orbit are proportional to the cubes also fill space according to matching rules was an innkeeper who collected herbs and made| could only be on ’s orbit—embodying his tough- 17th century “Theory of Everything9 ” in potions she believed had magical powers. At age 74, when she was of the average distance from the sun. just as Penrose tiles can cover a plane tried as a witch, Johannes acted as her lawyer and eventually won God at work. est and richest data—because (as Kepler which Kepler explains the math, but also her release at least partly because of ethical objections due to the surface quasiperiodically. authorities’ failure to follow correct legal procedures in the use of torture. 10 discovered) it’s most elliptical. He published his first two laws in Astro- mechanics and “music” of the universe. 2 By the end of his life, Kepler was owed some 12,000 gulden (about Kepler tested his $100,000 at current silver prices) by various rulers who were perpetually nomia nova (New Astronomy) in 1609, but Kepler Blocks broke. He died while trying to collect a debt. theory using the A year later, Tycho died of overindulgence. 3 didn’t articulate the 3rd until his 2nd mag- One of Kepler’s most interesting tilings is Maulbronn was haunted by the ill-famed Dr. Faustus a half-century earlier planetary Kepler had said Tycho was “superlatively 4 The system was funded by the Dukes of Württemburg to supply num opus, Harmonices mundi (Music of based on the number 5 (the “red” plane orbits deduced rich, but knows not how use it…one must educated clergy to defend the faith in religious battles raging between the Spheres) was already on press in 1619. in Zometool). Assembled from pentagons, Catholics and Protestants. by Copernicus, try to wrest his riches [i.e., his data] from 5 Tycho Brahe lost a chunk of his nose in a duel with another danish noble This contained a more elaborate model of decagons and stars, it doesn’t repeat A B C man, Manderup Parsberg. and the error him.” Now he did just that, as Tycho’s 6 the , and a goldmine of 2- and 3-D indefinitely nor articulate matching rules, Ironically, Kepler was excommunicated from the Lutheran Church in 1612, was less than heirs rushed to liquidate the estate. (Did Kepler blocks (A), assembled to form a rhombic a wound from which he never recovered. but it heralded Richert and Penrose tilings triacontahedron (B), Zometool parts removed (C) 7 Heinrich Kepler was a soldier of fortune: “a man viscous, inflexible, 10% (spectacular for a cosmological he inherit his dad’s mercenary slant?)7 Illustration of Kepler’s of the 20th century. quarrelsome and doomed to a bad end.” He abandoned the family when Johannes was 5 and presumably died in battle. model even today). He published his 2nd Law: a Kepler blocks made a startling reap- 8 planet sweeps 12 of the 13 Archimedean solids can be built with Zometool (although theory in the modestly titled Mysterium pearance in the 20th century with Daniel they require blue green lines, which we don’t consider part of the Zome War with Mars out equal system). Number 13, the , can be approximated. Cosmographicum (The Secret of the Shechtman’s discovery of quasicrystals. areas in equal 9 A. Garrett Lisi’s 21st century Exceptionally Simple Theory of Everthing Universe), blamed the discrepancies on Kepler had bet Tycho he could solve times (image A B C At the time, crystalography was consid- is based on an 8-dimensional space (E8) that can also be modeled using Zometool. Mars’s orbit in eight days; in fact, his exaggerates errors in Copernicus’s data... and sought the elipse’s Kepler tiling (A) with fat and skinny Richert diamonds ered a “closed” science and 5-fold crys- 10 See quote from Harmonices Mundi in the inset on the other side better data. “war with Mars” took him 8 years and a eccentricity.) in yellow, Richert tiling (B) and Penrose tiling (C) tals were “known” to be impossible. 11 D. Mermin, Physical Review Letters 68, 1172 (1992)

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for allages! and bubble-ology More great bubbles Crazy Bubbles by SteveBaer, ZomeworksCorp.,USA.©2009ZometoolInc. Contact [email protected] design; PaulHildebrandt,copywritingandproject management. Dr. Scott Vorthmann, vZome software for images; Anni Wildung, graphic Zometool Kepler’s ObsessionProject —Dr. JohnConway, concept; same: understandingouramazinguniverse. mechanics, the destination is always the to nanotechnology, quasicrystals to quantum ticket to discovery and fun. From numeracy Zometool answers, is your better learn or Whether youwanttoaskbetterquestions visit * together, wecanmaketheworldbetter. up whenyou’re done.Ifwework It’s alwaysagoodideatoclean than youfoundit. Leavetheplacecleaner with your fingers and pushing the ball straight off Take Zometoolmodelsapartbygraspingastrut you theshapeofstrutthatfits! lining uptheballsandlookingthroughholes.Theholesshow Hint: you can tell which strut fits between two balls in a model by models are alwaysstraight,neverundertension. a struttofititintotightspot,butstrutsinfinished Don’t force Zometool components. You can bend …and ifitdoesn’t work,itdoesn’t workatall. Ifitworks,worksperfectly. ZOMETOOL RULES! 3 2 1 We accidentallybrokenpartsforfree: replace zome fordetails. www.zometool.com/warranty Don’t breakitapart;takeapart! Related ProjectKits: or 303-297-3387 call 888-966-3386 zometool.com or Discover more at • • • Our mission: parts tobreak! or crushing them can cause balls, pullingmodelsapart with your thumb. Twisting buildabetterworld create value fun makelearning their duals. all 5PlaticSolidsand about andbuild Learn Platos Solids tool * .

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Euler Solids

Shadows Bubbles Plato’s 5 1 F 3. Useboard 2. Pointstrut 1. Placestrut + light rays. at 90ºto at thesun. in anyhole. V = E + 2 * 1 Tetrahedron 4 + 2 4 = 6 2 + 2 3 6 1 2 6 + Cube 8 = 12 3 3 + 2 7 4 1 Octahedron 2 8 + 6 4 = 12 3 + 2 4 1 8 Dodecahedron 2 12 + 20 20 = 3 30 30 + 2 4 “ the immeasurableexpanseoftime. doing so,setsitsvariousmarksonto pointsofcompletion;andby termined - dissonances), towardcertainprede uses these,inimitationofthosenatural syncopations andcadences(asMan through dissonanttensions,asifby the ear);amusic,whichprogresses music (perceivedbythemind,not else thanacontinuing,polyphonic thusnothing motionsare The planetary 20 20 Icoshedron + 12 – JohannesKepler, = 1619 Harmonices mundi, 30 30 + 2 of edges. Try it! Try edges. of number the than more two exactly is together faces and vertices of number the dron, polyhe- convex any for states * Euler’s formula Euler’s ”