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!

 

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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 [nucl-th] 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)

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Alan's Gyroid Notebooks

Uploaded to Commemorate His Passing on July 26, 2023

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OUTLINE OF TOPICS

• A. Polyhedra
  • 0. A random polyhedral honeycomb

             

  • 1. Roundest Polyhedra

    
    

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• B. Finite tilings by rhombs
  • 1. Rosettes and pseudo-rosettes

    

  • 2. ROMBIX

    ROMBIX is a [once-upon-a-time patented] 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.

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• C. Infinite tilings by rhombs
  • 1. Penrose tilings and pseudo-Penrose tilings

    

  • 2. d7-symmetric generalized Penrose tilings derived from de Bruijn heptagrids

    

  • 3. RPn tilings (recursive pseudo-Penrose tilings of dn symmetry)

    

  • 4. Rhombic wallpaper (periodic tilings derived from a variant form of the de Bruijn multigrid)

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• 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                     

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• E. Triply Periodic Minimal Surfaces (TPMS)
  • 0. Mathematical Preliminaries

    

  • 1. The P-G-D surface family

    

    

    

    The Yellow Moon Gyroid was installed on August 27, 2022  on the third floor Rotunda at the Morris Library, SIU Carbondale, Illinois.
    Click images for more information and video.

       

  • 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

    

    

   

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  • F. Other mathematicians, physicists, and chemists
    

    

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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)