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ALAN SCHOEN GEOMETRY The theory of roundest polyhedra (sometimes called best polyhedra ) was first treated by Simon Antoine Jean L'Huilier (1750-1840). One. of several accomplishments for which L'Huilier is remembered today. is his generalization to non-convex polyhedra of the famous formula. of Euler for convex polyhedra, which Euler first described in 1750 in. ALAN SCHOEN GEOMETRY In a 1970 NASA Technical Note Infinite Periodic Minimal Surfaces Without Self-Intersections (p.38 ff), I described how skeletal graphs can be used to represent TPMS. More recently David Hoffman and Jim Hoffman (no relation) have demonstrated in their Scientific Graphics Project that for the TPMS P, G, D, and also for a fourth surface (I-WP) of genus 4, there is a striking connection between ALAN SCHOEN GEOMETRY B1. ROSETTES (and PSEUDO-ROSETTES) . A rosette is an example of a finite tiling by rhombs. In § C1, C2, and C3, I consider a different kind of finite tiling by rhombs—the finite core of an infinite recursive tiling. A major difference between these two types of tiling is that while the number of rhombs of each shape in a rosette is fixed by the choice of the positive integer n, the number ALAN SCHOEN GEOMETRY The astonishing discovery of quasicrystals in 1984 by Dan Shechtman and his co-workers changed all that. Besides the extensive experiments on the growing of solid quasicrystals from the molten state, there has been much theoretical research — especially by Steinhardt and his collaborators, by Michael S. Longuet-Higgins, and by others too numerous to mention here — aimed at explaining the A CONSTRUCTION METHOD FOR TRIPLY PERIODIC MINIMAL 82 S. FUJIMORI AND M. WEBER Figure 2. A periodic polygon Because both boundary curves are invariant under a translation, we see that ∑m i=1 (ˇ − i) = 0 =∑n j=1 (ˇ − j): (1)Let P be a periodic polygon invariant under V.Let d=2 be the modulus of the annulus P= V , and define the strip Z = {z ∈ C : 0 < Imz < d=2}.The choice of d makes the annuli Z= h and P= V conformally equivalentTECHNICAL NOTE NASA
primitive cubic lattice. The Bravais lattice (lattice of trans- lational symmetry) for D is face-centered-cubic (F); the Bravais lattice for P is the primitive cubic lattice (P). A fundamental region of P or D is of genus 3. Both D and P are free of self-intersections. (It is readily shown* that FIGURE 1: “SIX EDGES” MINIMAL SURFACE Gergonne-Schwarz Surface 2 Define ρ0 = K = 1.6857503548 and let t= E(ξ) denote the functional inverse of the elliptic integral ξ= Zt 0 dτ √ 1+τ2 +τ4 The desiredGYRING GYROID
Gyring Gyroid The proposed sculpture is a spherical portion of the famous gyroid, a minimal surface found by Alan Schoen in 1970. The piece is to consist of 42 steel units; it is scaled to the geometry of its companion piece, Double Triamond, w/ Hexastix! WWW.SCHOENGEOMETRY.COM Created Date: 3/25/2013 5:47:26 AM ALAN SCHOEN GEOMETRY Welcome to the. GEOMETRY GARRET! A Pot-Pourri of People, Pictures, Places, Penrose Patterns, Polyhedra, Polyominoes, Posters, Posies, and Puzzles. Alan H. Schoen. Comments are welcome! It twists and turns throughout R3 in perfect helices. This net has two varieties – to left or right they spiral. When two are intertwined the combinationisn
ALAN SCHOEN GEOMETRY The theory of roundest polyhedra (sometimes called best polyhedra ) was first treated by Simon Antoine Jean L'Huilier (1750-1840). One. of several accomplishments for which L'Huilier is remembered today. is his generalization to non-convex polyhedra of the famous formula. of Euler for convex polyhedra, which Euler first described in 1750 in. ALAN SCHOEN GEOMETRY In a 1970 NASA Technical Note Infinite Periodic Minimal Surfaces Without Self-Intersections (p.38 ff), I described how skeletal graphs can be used to represent TPMS. More recently David Hoffman and Jim Hoffman (no relation) have demonstrated in their Scientific Graphics Project that for the TPMS P, G, D, and also for a fourth surface (I-WP) of genus 4, there is a striking connection between ALAN SCHOEN GEOMETRY B1. ROSETTES (and PSEUDO-ROSETTES) . A rosette is an example of a finite tiling by rhombs. In § C1, C2, and C3, I consider a different kind of finite tiling by rhombs—the finite core of an infinite recursive tiling. A major difference between these two types of tiling is that while the number of rhombs of each shape in a rosette is fixed by the choice of the positive integer n, the number ALAN SCHOEN GEOMETRY The astonishing discovery of quasicrystals in 1984 by Dan Shechtman and his co-workers changed all that. Besides the extensive experiments on the growing of solid quasicrystals from the molten state, there has been much theoretical research — especially by Steinhardt and his collaborators, by Michael S. Longuet-Higgins, and by others too numerous to mention here — aimed at explaining the A CONSTRUCTION METHOD FOR TRIPLY PERIODIC MINIMAL 82 S. FUJIMORI AND M. WEBER Figure 2. A periodic polygon Because both boundary curves are invariant under a translation, we see that ∑m i=1 (ˇ − i) = 0 =∑n j=1 (ˇ − j): (1)Let P be a periodic polygon invariant under V.Let d=2 be the modulus of the annulus P= V , and define the strip Z = {z ∈ C : 0 < Imz < d=2}.The choice of d makes the annuli Z= h and P= V conformally equivalentTECHNICAL NOTE NASA
primitive cubic lattice. The Bravais lattice (lattice of trans- lational symmetry) for D is face-centered-cubic (F); the Bravais lattice for P is the primitive cubic lattice (P). A fundamental region of P or D is of genus 3. Both D and P are free of self-intersections. (It is readily shown* that FIGURE 1: “SIX EDGES” MINIMAL SURFACE Gergonne-Schwarz Surface 2 Define ρ0 = K = 1.6857503548 and let t= E(ξ) denote the functional inverse of the elliptic integral ξ= Zt 0 dτ √ 1+τ2 +τ4 The desiredGYRING GYROID
Gyring Gyroid The proposed sculpture is a spherical portion of the famous gyroid, a minimal surface found by Alan Schoen in 1970. The piece is to consist of 42 steel units; it is scaled to the geometry of its companion piece, Double Triamond, w/ Hexastix! WWW.SCHOENGEOMETRY.COM Created Date: 3/25/2013 5:47:26 AM ALAN SCHOEN GEOMETRY Alan H. Schoen. Comments are welcome! A1. Roundest Polyhedra. is composed of one heptagon (green), fourteen pentagons (yellow), and eighteen hexagons (red-orange). 29 of the 32 faces of the 'soccer ball' polyhedron ( truncated icosahedron ). To view a sequence of LOMINOES - SCHOENGEOMETRY.COM 3 A2.FENCES A FENCE is a circuit, free of self-intersections, composed of LOMINOES laid end-to-end. A FENCE is called self-avoiding if every piece is incident only on the two pieces at its ends, and non-self-avoiding otherwise. The self-avoiding FENCE at the right is composed of all the pieces of L8†.The area enclosed by this fence isequal to 471.
K-PATTERNS - ALAN SCHOEN GEOMETRY D. K-PATTERNS: IMAGES DERIVED FROM PARTIAL SUMS OF POWER RESIDUES (STILL UNDER CONSTRUCTION!) BACKGROUND. We define a K-pattern as a chain of ν unit vectors u k, each of which joins r k and r k +1 (k = 0,1,2,), a consecutive pair of points in the plane, where The five parameters of a K-pattern — n, α, σ, ν, j 0 — are defined asfollows:
LOMINOES - SCHOEN GEOMETRY PREFACE LOMINOES are -Lshaped polyominoes composed of unit cubes. Tiling and packing properties of both the standard 28-piece set L8 and a 32-piece augmented set L8† are emphasized here, but both larger and smaller sets are also described. WWW.SCHOENGEOMETRY.COM Created Date: 3/25/2013 5:47:26 AMKEYST TONOES.
Gettingstarted Each ofthe sixteen pieces inaROMBlXset iscalled arombik. Fourofthe rombik arecalled keysttonoes. E•a•ch keystone isrhomb-shaped. The other twelve rombiks aretwins, made bycombining keystone shapes inpairs inevery possible way~to~Jformconcave hexagons. lJEDJ~~ ~ The rombiks belong tofourmonochrome subsets: green, yellow,red,blue. The bluesubset iscalled awkward; each ALAN SCHOEN GEOMETRY Created Date: 1/12/2011 12:05:56 PM ALAN SCHOEN GEOMETRY Created Date: 2/23/2009 12:24:05 PM ALAN SCHOEN GEOMETRY Created Date: 6/26/2011 1:34:16 PM SUM OF SINE IDENTITIES THE CASE Sum of Sine Identities The idea is: your algorithm does not produce all sum-of-sine identities but it seems to produce a spanning set for them. The case n =15. The cyclotomic polynomial of order 30 is: ALAN SCHOEN GEOMETRY B. Finite tilings by rhombs 1. Rosettes and pseudo-rosettes. 2. ROMBIX. ROMBIX is a puzzle that occurred to me unexpectedly while I was learning ALAN SCHOEN GEOMETRY A1. Roundest Polyhedra . The theory of roundest polyhedra (sometimes called best polyhedra) was first treated by Simon Antoine Jean L'Huilier (1750-1840). One of several accomplishments for which L'Huilier is remembered today is his generalization to non-convex polyhedra of the famous formula ALAN SCHOEN GEOMETRY In a 1970 NASA Technical Note Infinite Periodic Minimal Surfaces Without Self-Intersections (p.38 ff), I described how skeletal graphs can be used to represent TPMS. More recently David Hoffman and Jim Hoffman (no relation) have demonstrated in their Scientific Graphics Project that for the TPMS P, G, D, and also for a fourth surface (I-WP) of genus 4, there is a striking connection between ALAN SCHOEN GEOMETRY The astonishing discovery of quasicrystals in 1984 by Dan Shechtman and his co-workers changed all that. Besides the extensive experiments on the growing of solid quasicrystals from the molten state, there has been much theoretical research — especially by Steinhardt and his collaborators, by Michael S. Longuet-Higgins, and by others too numerous to mention here — aimed at explaining the ALAN SCHOEN GEOMETRY B1. ROSETTES (and PSEUDO-ROSETTES) . A rosette is an example of a finite tiling by rhombs. In § C1, C2, and C3, I consider a different kind of finite tiling by rhombs—the finite core of an infinite recursive tiling. A major difference between these two types of tiling is that while the number of rhombs of each shape in a rosette is fixed by the choice of the positive integer n, the number A CONSTRUCTION METHOD FOR TRIPLY PERIODIC MINIMAL 82 S. FUJIMORI AND M. WEBER Figure 2. A periodic polygon Because both boundary curves are invariant under a translation, we see that ∑m i=1 (ˇ − i) = 0 =∑n j=1 (ˇ − j): (1)Let P be a periodic polygon invariant under V.Let d=2 be the modulus of the annulus P= V , and define the strip Z = {z ∈ C : 0 < Imz < d=2}.The choice of d makes the annuli Z= h and P= V conformally equivalentTECHNICAL NOTE NASA
primitive cubic lattice. The Bravais lattice (lattice of trans- lational symmetry) for D is face-centered-cubic (F); the Bravais lattice for P is the primitive cubic lattice (P). A fundamental region of P or D is of genus 3. Both D and P are free of self-intersections. (It is readily shown* that FIGURE 1: “SIX EDGES” MINIMAL SURFACE Gergonne-Schwarz Surface 2 Define ρ0 = K = 1.6857503548 and let t= E(ξ) denote the functional inverse of the elliptic integral ξ= Zt 0 dτ √ 1+τ2 +τ4 The desired ALAN SCHOEN GEOMETRY Created Date: 6/26/2011 1:34:16 PM WWW.SCHOENGEOMETRY.COM Created Date: 3/25/2013 5:47:26 AM ALAN SCHOEN GEOMETRY B. Finite tilings by rhombs 1. Rosettes and pseudo-rosettes. 2. ROMBIX. ROMBIX is a puzzle that occurred to me unexpectedly while I was learning ALAN SCHOEN GEOMETRY A1. Roundest Polyhedra . The theory of roundest polyhedra (sometimes called best polyhedra) was first treated by Simon Antoine Jean L'Huilier (1750-1840). One of several accomplishments for which L'Huilier is remembered today is his generalization to non-convex polyhedra of the famous formula ALAN SCHOEN GEOMETRY In a 1970 NASA Technical Note Infinite Periodic Minimal Surfaces Without Self-Intersections (p.38 ff), I described how skeletal graphs can be used to represent TPMS. More recently David Hoffman and Jim Hoffman (no relation) have demonstrated in their Scientific Graphics Project that for the TPMS P, G, D, and also for a fourth surface (I-WP) of genus 4, there is a striking connection between ALAN SCHOEN GEOMETRY The astonishing discovery of quasicrystals in 1984 by Dan Shechtman and his co-workers changed all that. Besides the extensive experiments on the growing of solid quasicrystals from the molten state, there has been much theoretical research — especially by Steinhardt and his collaborators, by Michael S. Longuet-Higgins, and by others too numerous to mention here — aimed at explaining the ALAN SCHOEN GEOMETRY B1. ROSETTES (and PSEUDO-ROSETTES) . A rosette is an example of a finite tiling by rhombs. In § C1, C2, and C3, I consider a different kind of finite tiling by rhombs—the finite core of an infinite recursive tiling. A major difference between these two types of tiling is that while the number of rhombs of each shape in a rosette is fixed by the choice of the positive integer n, the number A CONSTRUCTION METHOD FOR TRIPLY PERIODIC MINIMAL 82 S. FUJIMORI AND M. WEBER Figure 2. A periodic polygon Because both boundary curves are invariant under a translation, we see that ∑m i=1 (ˇ − i) = 0 =∑n j=1 (ˇ − j): (1)Let P be a periodic polygon invariant under V.Let d=2 be the modulus of the annulus P= V , and define the strip Z = {z ∈ C : 0 < Imz < d=2}.The choice of d makes the annuli Z= h and P= V conformally equivalentTECHNICAL NOTE NASA
primitive cubic lattice. The Bravais lattice (lattice of trans- lational symmetry) for D is face-centered-cubic (F); the Bravais lattice for P is the primitive cubic lattice (P). A fundamental region of P or D is of genus 3. Both D and P are free of self-intersections. (It is readily shown* that FIGURE 1: “SIX EDGES” MINIMAL SURFACE Gergonne-Schwarz Surface 2 Define ρ0 = K = 1.6857503548 and let t= E(ξ) denote the functional inverse of the elliptic integral ξ= Zt 0 dτ √ 1+τ2 +τ4 The desired ALAN SCHOEN GEOMETRY Created Date: 6/26/2011 1:34:16 PM WWW.SCHOENGEOMETRY.COM Created Date: 3/25/2013 5:47:26 AM ALAN SCHOEN GEOMETRY The polyhedron P33, which I have conjectured is the roundest polyhedron with 33 faces, is composed of one heptagon (green), fourteen pentagons (yellow), and eighteen hexagons (red-orange). 29 of the 33 faces of P33 define a connected assembly with the same combinatorial structure as 29 of the 32 faces of the 'soccer ball' polyhedron (truncated icosahedron). ALAN SCHOEN GEOMETRY F. OTHER MATHEMATICIANS AND PHYSICISTS. Benoit Mandelbrot (left) died on October 14, 2010.. This photo was taken on April 29, 2006. Prof.Mandelbrot had
EXPANDABLE SPACE-FRAMES Title: EXPANDABLE SPACE-FRAMES Author: Alan H. Schoen, Alan H. Schoen Created Date: 10/16/2011 11:28:26 AM ALAN SCHOEN GEOMETRY Created Date: 2/19/2009 5:53:01 PM ALAN SCHOEN GEOMETRY Created Date: 1/12/2011 12:05:56 PM ALAN SCHOEN GEOMETRY Created Date: 2/23/2009 12:24:05 PM WWW.SCHOENGEOMETRY.COM Created Date: 3/25/2013 5:47:26 AM ALAN SCHOEN GEOMETRY Created Date: 11/9/2007 10:38:59 AM HIGH GENUS PERIODIC GYROID SURFACES OF NONPOSITIVE VOLUME 76, NUMBER 15 PHYSICAL REVIEW LETTERS 8APRIL 1996 High Genus Periodic Gyroid Surfaces of Nonpositive Gaussian Curvature Wojciech Gòz`dz` and Robert Hołyst Institute of Physical Chemistry and College of Sciences, Polish Academy of Sciences, Department III,Kasprzaka 44/52,
A F WELLS ORAL HISTORY A.M.S.tUEEN Wells (with pipe) on a transatlantic crossing on the Queen Mary to attend a crystallographic conference. Probably late 40s orearly 50s.
> WELCOME TO THE
> GEOMETRY GARRET!>
> A POT-POURRI OF PEOPLE, PICTURES, PLACES, PENROSE PATTERNS, > POLYHEDRA, POLYOMINOES, POSTERS, POSIES, AND PUZZLES>
> Alan H. Schoen
>
> _COMMENTS ARE WELCOME!_>
>
>
>
> ------------------------->
> Gyroid Jingle
> (Anon.)
>
> Fritz Laves found a crystal net with edges joined by threes, > It twists and turns throughout R3 in perfect helices. > This net has two varieties – to left or right they spiral. > When two are intertwined the combination isn’t chiral.>
> ‘Twixt two such nets a curving surface wends its way through> space,
> With tunnels everywhere that make it look just like old lace. > Now some do say the cosmos uses pasta (1) as its model. > (If that’s too big a stretch for you, you could just say it’s> twaddle.)
>
> R. Wagner wrote some famous operas, staging them at Bayreuth > (A name with several rhyming words – there’s thyroid and > there’s gyroid). > And that’s the name by which we know this anticlastic surface, > Whose labyrinths are such a maze they’re bound to make you> nervous.
>
> We hear that gyroid shapes are formed in heav’n in some stars. > (There’s no report that this occurs on Venus or on Mars.) > Such stars are not the common types like Capricorn or Castor, > But rather they’re like neutron stars (which spin around much> faster).
>
> Since many words do rhyme with ‘G’, like brie and ghee and plea, > To simplify this verse we’ll call the gyroid simply G. > Schwarz, Weierstrass, and Riemann taught us long ago to see > That G’s the offspring of two others known as P and D.>
> There’s Single G and Double G. Which one do you like more? > Since both occur in Nature, there’s no point in keeping score. > (2) Luzzati found that Double G can crystallize as soap, > And in the soap domain he’s known world over as the Pope.>
> Cosmologists say the universe arose by chance – not purpose. > So we conclude the gyroid’s just an accidental surface. > Vittorio2 found its structure hidden in a plain detergent. > And now phenomena like this are properly called emergent. > 1. Appearance of the Single Gyroid Network Phase in Nuclear Pasta> Matter,
> arXiv:1404.4760v5 31 Oct 2014, B. Schuetrumpf, M. A.> Klatt,
> K. Iida, G. E. Schroeder-Turk, J. A. Maruhn, K. Mecke, P.-G.> Reinhard
> 2. Luzzati, V. and Spegt, P. A., Nature, 215, 701 (1967)>
>
> OUTLINE OF TOPICS>
>> * A. Polyhedra>>
>> * 0. A RANDOM POLYHEDRAL HONEYCOMB>>
>>
>> * 1. ROUNDEST POLYHEDRA>>
>> ------------------------->>
>> * B. Finite tilings by rhombs>>
>> * 1. ROSETTES AND PSEUDO-ROSETTES>>
>> * 2. ROMBIX >> ROMBIX is a puzzle that occurred to me >> unexpectedly while I was learning >> about Penrose tilings from Martin Gardner's >> sensational January 1977 Scientific American article>> .
>>
>>
>> Here is a 42-page booklet >> that describes some of the variety of _COMBINATORIAL PROPERTIES_ >> of all ROMBIX sets.>>
>> These are the three versions of the ROMBIX puzzle currently >> available from KADON>>
>> The scrambled pieces of the ROMBIX-12 set at the extreme right are >> arranged in what I call a _chaotic_ tiling, >> but — just as for the two sets at its left — the pieces can be >> arranged very simply in an _orderly_ CRACKED EGG pattern.>>
>> Of these three ROMBIX sets, the one young children seem to enjoy>> most is
>> the _compound_ one at the left, which is composed of four _single_ >> sets of ROMBIX-4 . It was>> rediscovered
>> independently by Kate Jones (founder of KADON), who calls it>> 'ROMBIX Jr.'
>>
>> The most _versatile_ of these three sets is the one in the middle, >> the 16-piece ROMBIX-8 . >> (It's at the bottom of the KADON page, just below ROMBIX Jr.)>>
>> The most _challenging_ of these three sets is the one at the >> right, the 36-piece ROMBIX-12>> .
>> One of my favorite challenges with this set is the following: >> 1. Choose any one of the six colored subsets of rombiks. >> 2. Arrange the six rombiks in this subset in a tiling of the >> _central ladder_. >> 3. Fill in the rest of the tray with the remaining thirty rombiks >> to complete a _circle tiling_.>>
>> For those versed in combinatorics who wish to learn more about the >> fascinating combinatorial properties >> of _OVALS_ (convex polygons tiled by one or more rombiks), please >> see the journal article >> "Rhombic tilings of (_n,k_)-Ovals, (_n,k,λ_)-cyclic difference >> sets, and related topics" >> by my colleague John P. McSorley and me, which we published in >> 2013 in _Discrete Mathematics_.>>
>> ------------------------->>
>> * C. Infinite tilings by rhombs>>
>> * 1. PENROSE TILINGS AND PSEUDO-PENROSE TILINGS>>
>> * 2. _D7_-SYMMETRIC GENERALIZED PENROSE TILINGS DERIVED FROM DE >> BRUIJN HEPTAGRIDS>>
>> * 3. RP_N_ TILINGS (recursive pseudo-Penrose tilings of _dn_>> symmetry)
>>
>> * 4. RHOMBIC WALLPAPER (periodic tilings derived from a variant >> form of the de Bruijn multigrid)>>
>> ------------------------->>
>> * D. K-PATTERNS — aka "_RESI-DOODLES_": >> Images derived from partial sums of power and >> polynomial _RESIDUES_>>
>> 1 7 >> 11 >> 13 17 >> 19 23>>
>>
>> 29 31 >> 37 41 >> 43 >> 47 49>>
>>
>> 53 >> 59 61 >> 67 71 >> 73>>
>> 1 7 >> 11 >> 13 17 >> 19 23>>
>>
>> 29 31 >> 37 41 >> 43 >> 47 49>>
>>
>> 53 >> 59 61 >> 67 71 >> 73>>
>> 2 >> 4 8 >> 14 16 >> 22>>
>> 26 28 >> 32 34 >> 38 >> 44 46>>
>>
>> 52 56 >> 58 62 >> 64 >> 68 74>>
>>
>> 2 >> 4 8 >> 14 16 >> 22>>
>> 26 28 >> 32 34 >> 38 >> 44 46>>
>>
>> 52 56 >> 58 62 >> 64 >> 68 74>>
>>
>> ------------------------->>
>> * E. Triply Periodic Minimal Surfaces (TPMS)>>
>> * 0. MATHEMATICAL PRELIMINARIES>>
>> * 1. THE _P-G-D_ SURFACE FAMILY>>
>> * 2. CUBIC LATTICE SURFACES NOT IN THE _P-G-D_ FAMILY>>
>> * 3. TRIANGLE LATTICE SURFACES>>
>> * 4. SURFACES ON OTHER LATTICES>>
>> * 5. BACKGROUND>>
>> * 6. BIBLIOGRAPHY>>
>> * 7. MINIMAL SURFACE PEOPLE>>
>>
>> ------------------------->>
>> * F. Other mathematicians, physicists, and chemists>>
>
> ------------------------->
> FUTURE STUFF
>
>> * 2-dimensional puzzles>>
>> * QUARKS>>
>> * LOMINOES>>
>> For an illustrated ten-page introductory BOOKLET about LOMINOES,>> look here ,
>> but for an encyclopedic BOOK (131 pages) about LOMINOES, look here>> .
>>
>> NOTE: On pp. 129 and 131 of the LOMINOES BOOK, the address listed >> as the link to Neil J. A. Sloane's >> ONLINE ENCYCLOPEDIA FOR INTEGER SEQUENCES is obsolete and should >> be replaced by http://oeis.org/ >> Be sure to watch (and hear) Tony Noe's spectacular 8.5 minute >> movie illustrating Sloane's Encyclopedia!>>
>> * 3-dimensional puzzles>>
>> * TETRONS, CUBONS, OCTONS, DODECONS, and ICONS>>
>> * INCUBUS cube puzzle>>
>> * OCTO (double set of the eight solid tetrominoes)>>
>> * STARBIX and >> other closed chains of polyhedra>>
>> * posters>>
>> * RHOMBBURST (4-color poster) >> * _H_, an embedded triply-periodic minimal surface parametrized >> by Hermann Amandus Schwarz in 1866 (b&w poster) >> * _F-RD_, an embedded triply-periodic minimal surface identified >> by the author in 1969 (b&w poster)>
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