Sources page biographical material

DF P P&dV D B J K

Truncated tetrahedron T,22v 4v,II-69v 3 35 56 A3 2

Truncated cube 22r 36 61 G2 1

Truncated dodecahedron 21r 76 3

Truncated octahedron 21v 17 38 68 B2,B4 5

Great rhombi-cubo-octahedron 41 88 B6 6

Great rhombi-icosidodecahedron 100 7

Truncated icosahedron (football!) 20v 4v 23 + 81 4

Cubo-octahedron T 4v,II-69v 9 37 58 B6,F1 8

Icosidodecahedron 4v 29 71 D4,F6 9

Rhombi-cubo-octahedron 35 39 64 B6 10

Rhombi-icosidodecahedron 94 D4 11

Snub cube 40 12

Snub dodecahedron 13

NUMBER 5(6) 4 6 7(9) 11 8 13
Richard Buckminster Fuller. Centre spread card version of his Dymaxion World map on the cubo-octahedron. Life (15 Mar 1943). Reproduced in colour, with extended discussion, in: Joachim Krausse & Claude Lichtenstein, eds; Your Private Sky R. Buckminster Fuller The Art of Design Science [book accompanying a travelling exhibition in 2000]; Lars Müller Publishers, Bade, Switzerland, 1999, pp. 250-275. (This quotes a Life article on 1 Mar 1943 and a Fuller article, Fluid Geography, of 1944 -- ??NYS. It also reproduces a 1952 colour example of the icosahedral version.)

Richard Buckminster Fuller. US Patent 2,393,676 -- Cartography. Filed: 25 Feb 1944; granted: 29 Jan 1946. 3pp + 5pp diagrams. His world map on the cubo-octahedron. It was later put on the icosahedron. One page is reproduced in: William Blackwell; Geometry in Architecture; Key Curriculum Press, Berkeley, 1984, p. 157.

J. H. Conway. Four-dimensional Archimedean polytopes. Proc. Colloq. Convexity, Copenhagen, 1965 (1967) 38-39. ??NYS -- cited by Guy, CMJ 13:5 (1982) 290-299.
6.AT.4. UNIFORM POLYHEDRA
H. S. M. Coxeter, M. S. Longuet Higgins & J. C. P. Miller. Uniform polyhedra. Philos. Trans. Roy. Soc. 246A (1954) 401 450. They sketch earlier work and present 53 uniform polyhedra, beyond the 5 Platonic, 13 Archimedean and 4 Kepler Poinsot polyhedra and the prisms and anti prisms. Three of these uniform polyhedra are actually infinite families. "... it is the authors' belief that the enumeration is complete, although a rigorous proof has still to be given."

S. P. Sopov. Proof of completeness of the list of uniform polyhedra. Ukrain. Geometr. Sb. 8 (1970) 139-156. ??NYS -- cited in Skilling, 1976.

J. S. Skilling. The complete set of uniform polyhedra. Philos. Trans. Roy. Soc. London Ser. A 278 (1975) 111 135. Demonstrates that the 1954 list of Coxeter, et al., is complete. If one permits more than two faces to meet at an edge, there is one further polyhedron -- the great disnub dirhombidodecahedron.

J. S. Skilling. Uniform compounds of uniform polyhedra. Math. Proc. Camb. Philos. Soc. 79 (1976) 447-468. ??NYS -- I am told it determines that there are 75 uniform compounds and also cites Sopov.

6.AT.5. REGULAR FACED POLYHEDRA
O. Rausenberger. Konvexe pseudoreguläre Polyeder. Zeitschr. für math. und naturwiss. Unterricht 46 (1915) 135 142. Finds the eight convex deltahedra.

H. Freudenthal & B. L. van der Waerden. Over een bewering van Euclides [On an assertion of Euclid] [in Dutch]. Simon Stevin 25 (1946/47) 115 121. ??NYS. Finds the eight convex deltahedra -- ignorant of Rausenberger's work.

H. Martyn Cundy. "Deltahedra". MG 36 (No. 318) (Dec 1952) 263 266. Suggests the name "deltahedra". Exposits the work of Freudenthal and van der Waerden, but is ignorant of Rausenberger. Considers non convex cases with two types of vertex and finds only 17 of them. Considers the duals of Brückner's trigonal polyhedra.

Norman W. Johnson. Convex polyhedra with regular faces. Canad. J. Math. 18 (1966) 169 200. (Possibly identical with an identically titled set of lecture notes at Carleton College, 1961, ??NYS.) Lists 92 such polyhedra beyond the 5 regular and 13 Archimedean polyhedra and the prisms and antiprisms.

Viktor A. Zalgaller. Convex polyhedra with regular faces [in Russian]. Seminars in Mathematics, V. A. Steklov Mathematical Institute, Leningrad, vol. 2 (1967). ??NYS. English translation: Consultants Bureau, NY, 1969, 95pp. Gives details of computer calculations which show that Johnson's list is complete. Defines a notion of simplicity and shows that the simple regular faced polyhedra are the prisms, the antiprisms (excepting the octahedron) and 28 others. Names all the polyhedra and gives drawings of the simple ones.
6.AT.6. TESSELLATIONS
Albrecht Dürer. Underweysung der messung .... 1525 & 1538. Op. cit. in 6.AA. Figures 22 27 (pp. 156 169 in The Painter's Manual, Dürer's 1525 ff. E vi v   F-iii-v) show: the three regular tessellations; the quasi-regular one, 3636, and some of its dual; several irregular ones, including some partial tessellations with pentagons; and the truncated square lattice, 48². In the revised version of 1538, he adds some tilings by rhombuses (figures 23a & 24, pp. 410 411 in The Painter's Manual).

Albrecht Dürer. Elementorum Geometricorum (?). Op. cit. in 6.AA, 1532. Book II, fig. 22 27, pp. 62-67, is the material from the 1525 version.

J. Kepler. Letter to Herwart von Hohenberg. 6 Aug 1599. Op. cit. in 6.AT.2. Field, p. 105, says Kepler discusses tessellations here and this is the earliest of his writings to do so.

J. Kepler. Harmonices Mundi. 1619. Book II. Opp. cit. above. Prop. XVIII, p. 51 & plate (missing in my facsimile). Shows there are only three regular plane tessellations and mentions the dual of 3636, which is the 'baby blocks' tessellation. Prop. XIX XX, pp. 51 56 & four plates (three missing in my facsimile). Finds the 8 further Archimedean tessellations and 7 of the 10 further ways to fill 360 degrees with corners of regular polygons. He misses 3,7,42; 3,8,24; 3,9,18 despite computing, e.g., that a triangle and a heptagon would leave a gap of 40/21 of a right angle. Field, p. 109, notes that Kepler doesn't clearly have all vertices the same in some pictures -- e.g. he has both 3366 and 3636 patterns in his figure R.

Koloman Moser. Ver Sacrum. 1902. This Viennese art nouveau drawing is considered to be the first tessellation using life-like figures. It has trout and the pattern has symmetry (or wallpaper) group pg and isohedral type IH2.

Branko Grünbaum & G. C. Shephard. Tilings and Patterns. Freeman, 1986. I haven't examined this thoroughly yet, but it clearly is the definitive work and describes everything known to date.

6.AT.6.a. TESSELLATING WITH CONGRUENT FIGURES
This is a popular topic which I have just added. Gardner's article and addendum in Time Travel gives most recent results, so I will just give just some highlights. The facts that any triangle and any quadrilateral will tile the plane must be very old, perhaps Greek, but I have no early references. Generally, I will consider convex polygons and most items only deal with the plane.
David Hilbert. Mathematische Probleme. Göttinger Nachrichten (= Nachrichten der K. Gesellschaft der Wissenschaften zu Göttingen, Math. phys. Klasse) 3 (1900) 253 297. This has been reprinted and translated many times, e.g. in the following.

R. Bellman, ed. A Collection of Modern Mathematical Classics -- Analysis. Dover, 1961. Pp. 248 292 [in German].

Translated by M. W. Newson. Bull. Amer. Math. Soc. 8 (1902) 437 479. Reprinted in: F. E. Browder, ed. Mathematical Developments Arising from Hilbert Problems. Proc. Symp. Pure Math. 28 (1976) 1 34.

Problem 18: Aufbau des Raumes aus kongruenten Polyedern [Building up of space from congruent polyhedra]. "The question arises: Whether polyhedra also exist which do not appear as fundamental regions of groups of motions, by means of which nevertheless by a suitable juxtaposition of congruent copies a complete filling up of space is possible." Hilbert also asks two other questions in this problem.

The problem is discussed by John Milnor in his contribution to the Symposium, but he only shows non convex 8  & 10 gons which fill the plane.

K. Reinhardt. Über die Zerlegung der Ebene in Polygone. Dissertation der Naturwiss. Fakultät, Univ. Frankfurt/Main, Borna, 1918. ??NYS -- cited by Kershner. Finds the three types of hexagons and the first five types of pentagons which fill the plane.

Max Black. Reported in: J. F. O'Donovan; Clear thinking; Eureka 1 (Jan 1939) 15 & 20. Problem 2: which quadrilaterals can tile the plane? Answer: all!

R. B. Kershner. On paving the plane. AMM 75:8 (Oct 1968) 839 844. Says the problem was posed by Hilbert. Gives exhaustive lists of hexagons and a list of pentagons which he claimed to be exhaustive. Cites previous works which had claimed to be exhaustive, but he has found three new types of pentagon.

J. A. Dunn. Tessellations with pentagons. MG 55 (No. 394) (Dec 1971) 366 369. Finds several types and asks if there are more.

M. M. Risueño, P. Nsanda Eba & Editorial comment by Douglas A. Quadling. Letters: Tessellations with pentagons. MG 56 (No. 398) (Dec 1972) 332 335. Risueño's letter replies to Dunn by citing Kershner. Eba constructs a re entrant pentagon. [This is not cited by Gardner.]

Gardner. On tessellating the plane with convex polygon tiles. SA (Jul 1975). Much extended in Time Travel, chap. 13.

Ivan Niven. Convex polygons that cannot tile the plane. AMM 85 (1978) 785-792. n gons, with n > 6, cannot tile the plane.

Doris Schattschneider. In praise of amateurs. In: The Mathematical Gardner; ed. by David A. Klarner; Wadsworth, Belmont, California, 1981, pp. 140 166 & colour plates I V between 166 & 167. Surveys history after Kershner, describing contributions of James & Rice.

Gardner. On tessellating the plane with convex polygon tiles. [Originally: SA (Jul 1975).] Much extended in Time Travel, 1988, chap. 13. The original article generated a number of responses giving new pentagonal tilings, making 14 types in all. Good survey of the recent literature.

6.AT.7. PLAITING OF POLYHEDRA
New section.
John Gorham. A System for the Construction of Crystal Models on the Type of an Ordinary Plait: Exemplified by the Forms Belonging to the Six Axial Systems in Crystallography. E. & F. N. Spon, London, 1888. Gorham's Preface says he developed the idea and demonstrated it to the Royal Society some 40 years earlier.

A. R. Pargeter. Plaited polyhedra. MG 43 (No. 344) (May 1959) 88 101. Cites and quotes Gorham. Extends to plaiting dodecahedron, icosahedron and some archimedean, dual and stellated examples.

J. Brunton. The plaited dodecahedron. MG 44 (No. 347) (Feb 1960) 12 14. With comment by Pargeter. Obtains a 3 plait which almost completes the dodecahedron.
6.AT.8. DÜRER'S OCTAHEDRON
New section -- I know of other articles claiming to 'solve' the problem.
Albrecht Dürer. Melencolia I. 1514. Two impressions are in the British Museum. In the back left is an octahedron whose exact shape is the subject of this section. It looks like a cube truncated at two opposite corners, but the angles do not quite look like 90^o.

Albrecht Dürer. Dresden Sketchbook. Facsimile as The Human Figure, the complete Dresden Sketchbook; Dover, NY, 1972. ??NYS -- cited by Sharp. This has a sketch of the solid with hidden lines indicated, so the combinatorial shape is definitely known and is a hexahedron of six equal faces, truncated at two opposite corners.

E. Schröder. Dürer Kunst und Geometrie. Birkhäuser, Basel, 1980. ??NYS -- cited by Sharp and MacGillavry.

Caroline H. MacGillavry. The polyhedron in A. Dürer's Melencolia I An over 450 years old puzzle solved? Koninklijke Nederlandse Akademie van Wetenschappen Proc. B 84:3 (28 Sep 1981) 287-294. The rhombohedral angle, i.e. the angle between edges at the truncated top and bottom vertices of the rhombohedron, was estimated as 72^o by Grodzinski. She determines it is 79^o ± 1^o. She then built and photographed such a polyhedron and then computed its projection, both of which seem identical to Dürer's picture. Crystallographers believe Dürer was drawing an actual crystal, with a form of calcite having rhombohedral angle of 76^o being the closest known shape, though it is not known to have been studied in Dürer's time, so others have suggested fluorite, though fluorite has two standard forms, neither of this form, but Dürer's 'hybrid' artistic version could have been derived from them.

Terence Lynch. The geometric body in Dürer's engraving Melencolia I. J. Warburg and Courtauld Institutes 45 (1982) 226-232 & plate a on p. 37?. Lots of references to earlier work. Notes that perspective was not sufficiently advanced for Dürer to construct a general drawing of such an object. After many trials, he observes that a parallel projection of the solid fits onto a 4 x 4 grid -- like the magic square in the picture -- and that symmetry then permits the construction with straight edge and compass (which are both shown in the picture). This shows that the original faces are rhombuses whose diameters are in the ratio 2 : 3. And the dihedral angle between the triangular faces and the cut off rhombuses is 30^o Further, the actual drawing can then be made by one of the simplified perspective techniques known to Dürer. However, Dürer has taken a little bit off the top and bottom of the figure and this distortion has misled many previous workers.

John Sharp. Dürer's melancholy octahedron. MiS (Sep 1994) 18-20. Asserts that the shape was first determined by Schröder in 1980 and verified by Lynch.

6.AT.9. OTHER POLYHEDRA
New section.
Stuart Robertson. The twenty-two cuboids. Mathematics Review 1:5 (May 1991) 18-21. This considers polyhedra with six quadrilateral faces and determines what symmetries are possible -- there are 22 different symmetry groups.
6.AU. THREE RABBITS, DEAD DOGS AND TRICK MULES
See S&B, p. 34.

Loyd's Trick Mules has two mules and two riders which can only be placed correctly by combining each front with the other rear.

Earlier forms showed two dead dogs which were brought to life by adding four lines. The resulting picture is a pattern, generally called 'Two heads, four children' and can be traced back to medieval Persian, Oriental and European forms.

The three rabbits problem is: "Draw three rabbits, so that each shall appear to have two ears, while, in fact, they have only three ears between them." Until about 1996, I only knew this from the 1857 Magician's Own Book and the many books which copied from it. Someone at a conference at Oxford in 1996 mentioned that the pattern occurs in a stained glass window at Long Melford, Suffolk. Correspondence revealed that the glass is possibly 15C and the pattern was apparently brought from Devon about that time. More specifically, it comes from the east side of Dartmoor and inquiries there have turned up numerous examples as roof bosses from 13 16C. Totally serendipitously, I was reading a guide book to Germany in 1997 and discovered the pattern occurs in stonework, possibly 16C, at Paderborn, Germany. A letter led to receiving a copy of Schneider's article (see below) which described the pattern occurring at Dunhuang, c600. I am indebted to Miss Y. Yasumara, the Art Librarian at the School of Oriental and African Studies, for directing me to several works on Dunhuang. However, I have not examined all these works in detail (the largest is five large volumes), so I may not have found all the examples of this pattern. Miss Yasumara also directed me to Roderick Whitfield, of the School of Oriental and African Studies, who tells me there is no other example of this pattern in Chinese art, and to Susan Whitfield (no relation), head of the International Dunhuang Project at the British Library. However, Greeves (see his articles, below) has found other examples of the pattern in Europe, Iran and Tibet and found that modern carpets with the pattern are being made in China. A student of his recently went to Dunhuang and the locals told her that the pattern came from 'the West', meaning India, which opens up a whole new culture to examine.

In 1997, I visited the Dartmoor area, seeing several examples and finding a reference (Hambling, below) to Tom Greeves' article. In 2000, I again visited the area, seeing more churches and meeting Tom Greeves and his associate Sue Andrew. Later in 2000, I visited Paderborn. Later in 2000, I showed this material to Wei Zhang and Peter Rasmussen, leading collectors of Chinese puzzles and they have begun to investigate the Chinese material much more thoroughly than I have done -- see below. In 2001, I went again to the area.

I have read that rabbits were introduced to England in 1176 by the Normans and became common in the 13C, though I believe they weren't really wild for some time after that. E.g. [J. A. R. Pimlott; Recreations; Studio Vista, 1968, p. 18] says: "The rabbit was not known in Britain until the thirteenth century and did not become plentiful until the fifteenth", citing Elspeth M. Veale; The English Fur Trade in the Later Middle Ages; 1966. I have read that hares were introduced between -500 and +500, but I have just seen a mention that bones of a hare found in Ireland have been carbon dated to  26,000. So it is probable that the animals in many of the images are hares, though some of the images are distinctly more rabbit-like than hare-like.

The material in this section has grown so much that it is now divided into seven subsections: China; Other Asia; Paderborn; Medieval Europe; Modern Versions of the Three Rabbits Puzzle; Dead Dogs; Trick Mules.

I have about a dozen letters and emails which have not yet been processed.

Rabbits going clockwise: Dunhuang (14 caves -- all except 407 & 420); Goepper; St. Petersburg; Iranian tray; Paderborn; Münster; Bestiary; Lyon; Throwleigh; Valentine; Clyst Honiton; Hasloch am Main; Michelstadt; Collins; Greeves letterhead; Urumqi;

Rabbits going anti-clockwise: Dunhuang (2 caves -- 407 & 420); Corbigny; Lombard's Gloss; North Bovey; Long Crendon; Chester; Widecombe; Long Melford; Chagford; Best Cellars, Chagford; Tavistock; Broadclyst; Sampford Courtenay; Spreyton; Paignton; South Tawton; Valentine; Schwäbisch Hall; Baltrušaitis Fig. 97; Child; Magician's Own Book et al; Warren Inn; Best Cellars, Chagford; Newman; Lydford; Trinity Construction Services;

The choice of clockwise versus anticlockwise seems to be random! Except the Chinese clearly preferred clockwise.

FOUR RABBITS versions. Baltrušaitis; Wilson; Goepper. Bestiary; Andrew/Lombard; Lyon; Hamann-MacLean;

MORE FIGURES. Chichester Cathedral. Boxgrove Priory.
CHINA
Jurgis Baltrušaitis. Le Moyen Age Fantastique Antiquités et exotismes dans l'art gothique. (A. Colin, Paris, 1955, 299 pp.) Revised, Flammarion, Paris, 1981, 281 pp. Thanks to Peter Rasmussen for telling me about this. Supposedly an English edition was published in 1998, but I have found an entry in the Warburg catalogue for The Fantastic in the Middle Ages, published by Boydell & Brewer, Woodbridge, Suffolk, 2000, marked 'order cancelled' -- so the publication seems to have never happened. This is a major source used for When Silk Was Gold, below. Pp. 132 139 of the 1981 edition have many examples of three and four rabbits, four boys, etc. He gives a small illustration (Fig. 96B) from Dunhuang (Touen-houang) (6C-10C), the oldest example he knows, and many others. See other sections for more details.

[Huang Zu'an ?? -- Schneider, below, gives this author, but there is no mention of an author in the entire issue.] Dunhuang -- Pearl of the Silk Road. China Pictorial (1980:3) 10-23 with colour photo on p. 22. 9th article in a series on the Silk Road. Colour photo of the three rabbits pattern with caption: "A ceiling design. The three rabbits with three ears and the apsarases seem to be whirling. Cave 407. Sui Dynasty." The Sui dynasty was from 581 to 618, so we can date this as c600. The image is rather small, but the three hares can be made out. There is no discussion of the pattern in the article.

Chang Shuhong & Li Chegxian. The Flying Devis of Dunhuang. China Travel and Tourism Press, Beijing, 1980. Unpaginated. In the Preface, we find the following. "What is particularly novel is the full-grown lotus flower painted in the centre of the canopy design on the ceiling of Cave 407. In the middle of the flower there are three rabbits running one after the other in a circle. For the three rabbits only three ears are painted, each of them borrowing one ear from another. This is an ingenious conception of the master painter." From this, it seems that this pattern is uncommon. The best picture of the pattern that I have located is in this book, in the section on the Sui period. I have now acquired a copy with its dust jacket and find a painting of the pattern is on the back of the dust jacket -- this is the painting also given in Li Kai et al, below, p, 27. [Incidentally, a devi or apsaras is a kind of Buddhist angel. The art of Dunhuang is quite lovely.]

The Dunhuang Institute for Cultural Relics. The Mogao Grottos of Dunhuang. 5 vols. + Supplement. Heibonsha Ltd., Tokyo, 1980-1982. (In Japanese, with all the captions given in English at the end of the Supplementary volume. Fortunately the plate and cave numbers are in western numerals. A Chinese edition was planned.) Vol. 2, plate 94, is a double-page spread of the ceiling of Cave 407, with the page division running right through the middle of the rabbits pattern! Vol. 2, plate 95, is a half-page plate of the ceiling of Cave 406, and shows the rabbits pattern, but it seems rather faded. The English captions simply say "Ornamental ceiling decoration".

R. Whitfield & A. Farrer. Caves of the Thousand Buddhas. British Museum, 1990, esp. pp. 12 & 16. Though cited by Greeves, these pages only have general material on Dunhuang and the book does not mention any of the relevant caves.

Duan Wenjie. Dunhuang Art Through the Eyes of Duan Wenjie. Indira Gandhi National Centre for the Arts, Abhinav Publications, New Delhi, 1994. This gives much more detail about the caves. Pp. 400-401 describes Caves 406-407. Peter Rasmussen examined the book in detail and found it mentions 12 other caves with the pattern. Peter has found that the book is accessible on-line at www.ignca.nic.in/ks_19.htm. This provides the facility to download a font which will display diacritical marks and this is worth doing before you start to browse.

In late 2002, Peter Rasmussen and Wei Zhang were able to inspect 10 caves that they hadn't seen before. Peter sent notes of their impressions of 12 caves on 24 Nov 2002 and I will add some of Peter's comments and additions in [ ] and marked PR. This will make the following list the basic list of all sixteen of the caves.
Cave No. 127. Late Tang (renovated in Five Dynasties and Qing). "The ceiling exhibits lotus and three rabbits (joining as one) in the centre." [PR: tan on turquoise, going clockwise.]

Cave No. 139. Late Tang. "The ceiling shows the three rabbits (joining as one) and lotus designs in the centre." [PR: this is a small cave off the entry to Cave 138. Tan on light green, going clockwise, in excellent condition. "Rabbits beautifully drawn in pen-like detail, showing toes, eyes (with eyeballs!), all four legs, tail, nose, mouth, outline of thigh muscle, and hair on stomach, breast, legs and top of head." Peter says this is by far the finest of the images; he is applying to get it photographed.]

Cave No. 144. Middle and Late Tang (renovated during the Five Dynasties and Qing). "The centre of the ceiling shows the three rabbits (joining as one) and floral designs." [PR: white on aqua green, going clockwise.]

Cave No. 145. Late Tang (renovated during the Five Dynasties and Song). "The ceiling of the niche on the west wall shows lotus and three rabbits (joining as one), chess board and floral patterns." [Zhang & Rasmussen's letter of 1 Jun 2001 says the rabbits are going clockwise.]

Cave No. 147. Late Tang. "Main Hall: The ceiling shows three rabbits (joining as one) and lotus designs in the centre". [PR: tan or turquoise green, going clockwise. Paint on ears has peeled off.]