Sources page biographical material

6.AS.1.a. GREEK CROSS TO A SQUARE
Note that a proper Greek cross is formed from five equal squares.
Lucas. RM2. 1883. Loc. cit. in 6.AS.1. Uses 20 triangles.

Lemon. 1890. The Maltese cross squared, no. 369, pp. 51 & 111. Cut a Maltese cross (really a Greek cross) by two cuts into four pieces that make a square.

Hoffmann. 1893. Chap. III, no. 13: The Greek cross puzzle, pp. 94 & 126 = Hoffmann Hordern, pp. 82 & 84. Has four pieces made by two cuts.

Loyd. Tit Bits 31 (10, 17 & 31 Oct 1896) 25, 39 & 75. = Cyclopedia, 1914, p. 14. Four pieces as in Hoffmann.

Loyd. Problem 23: A new "square and cross" puzzle. Tit Bits 31 (13 Mar 1897) 437 & 32 (3 Apr 1897) 3. = Cyclopedia, 1914, pp. 58, 270 & 376. Four congruent pieces.

Loyd. Problem 27: The swastika problem. Tit Bits 32 (3 & 24 Apr 1897) 3 & 59. = Cyclopedia, 1914, p. 58. Quadrisect square to make two equal Greek crosses.

Loyd. Problem 30: The Easter problem. Tit Bits 32 (24 Apr & 15 May 1897) 59 & 117. Dissect square into five pieces to make two unequal Greek crosses.

Dudeney. Problem 56: Two new cross puzzles. Tit Bits 33 (23 Oct & 13 Nov 1897) 59 & 119. Dissect a half square (formed by cutting a square either vertically or diagonally) to a Greek cross. Solutions in 3 and 4 pieces. [The first case = Loyd, Cyclopedia, 1914, Easter 1903, pp. 46 & 345.]

Benson. 1904. The Greek cross puzzle, p. 197. = Hoffmann, p. 94.

Dudeney. Cutting-out paper puzzles. Cassell's Magazine ?? (Dec 1909) 187-191 & 233-235.

States that the dissection with four pieces in two cuts is relatively 'recent'. c= AM, 1917, p. 29, which dates this to 'the middle of the nineteenth century'.

Fold a Greek cross so that one cut gives four congruent pieces which form a square. = AM, 1917, prob. 145, pp. 35 & 169.

M. Adams. Indoor Games. 1912. The Greek cross, p. 349 with figs. on p. 347.
6.AS.1.b. OTHER GREEK CROSS DISSECTIONS
See also 6.F.3 and 6.F.5.
Dudeney. A batch of puzzles. Royal Magazine 1:3 (Jan 1899) & 1:4 (Feb 1899) 368-372. Squares and cross puzzle. = AM, 1917, p. 34. Dissect a Greek cross into five pieces which make two squares, one three times the edge of the other. If the squares in the Greek cross have edge 2, then the cross has area 10 and the two squares have areas 1 and 9. The dissection arise by joining the midpoints of the edges of the central square of the cross and extending these lines in one direction symmetrically.

Dudeney. AM. 1917. Greek cross puzzles, pp. 28-35. This discusses a number of examples and gives a few problems.

Collins. Book of Puzzles. 1927. The Greek cross puzzle, pp. 98-100. Take a Greek cross whose squares have side 2, so the cross has area 10. Take another cross of area 5 and place it inside the large cross. If this is done centrally and the small one turned to meet the edges of the large one, there are four congruent heptagonal pieces surrounding the small one which make another Greek cross of area 5.

Eric Kenneway. More Magic Toys, Tricks and Illusions. Beaver Books (Arrow (Hutchinson)), London, 1985. On pp. 56-58, he considers a Greek cross cut by two pairs of parallel lines into nine pieces which would make five squares. The lines join an outer corner to the midpoint of an opposite segment. This produces a tilted square in the centre. By pairing the other pieces, he gets four identical pieces which make a square and a Greek cross in a square.

6.AS.2. TWO (ADJACENT) SQUARES TO A SQUARE
The smaller square often has half the edge of the larger, which connects this with 6.AS.1, but this is not essential. The two squares are usually viewed as one piece, i.e. a P pentomino. These items are dissection proofs of the Theorem of Pythagoras -- see Yates (op. cit. in 6.B, pp. 38-39) for some other examples of this point.

See Ripley's for a similar example, but the 2 x 2 square has a 2, 2, 2 triangle attached to an edge.

Another version has squares of area 1 and 8. The area 8 square is cut into four pieces which combine with the area 1 square to make an area 9 square. I call this the 4 - 5 piece square.
Walther Karl Julius Lietzmann (1880-1959). Der Pythagoreische Lehrsatz. Teubner, (1911, 2nd ed., 1917), 6th ed., 1951. [There was a 7th ed, 1953.] Pp. 23-24 gives the standard dissection proof for the Theorem of Pythagoras. The squares are adjacent and if considered as one piece, the dissection has three pieces. He says it was known to Indian mathematicians at the end of the 9C as the Bride's Chair (Stuhl der Braut). (I always thought this name referred to the figure of the Euclid I, 47 -- ??)

Thabit ibn Qurra (= Thābit ibn Qurra). c875. Gives the standard dissection proof for the Theorem of Pythagoras. The squares are adjacent and if considered as one piece, the dissection has three pieces. [Q. Mushtaq & A. L. Tan; Mathematics: The Islamic Legacy; Noor Publishing House, Farashkhan, Delhi, 1993, pp. 71-72] give this and cite Lietzmann. Greg Frederickson [email of 18 Oct 1996] cites Aydin Sayili; Thabit ibn Qurra's generalization of the Pythagorean theorem; Isis 51 (1960) 35-37.

Abu'l 'Abbas al-Fadhl ibn Hatim al-Narizi (or Annairizi). (d. c922.) Ed. by Maximilian Curtze, from a translation by Gherardo of Cremona, as: Anaritii In decem libros priores Elementorum Euclidis Commentarii, IN: Euclidis Opera Omnia; Supplementum; Teubner, Leipzig, 1899. ??NYS -- information supplied by Greg Frederickson.

Johann Christophorus Sturm. Mathesis Enumerata, 1695, ??NYS. Translated by J. Rogers? as: Mathesis Enumerata: or, the Elements of the Mathematicks; Robert Knaplock, London, 1700, ??NYS -- information provided by Greg Frederickson, email of 14 Jul 1995. Fig. 29 shows it clearly and he attributes it to Frans van Schooten (the Younger, who was the more important one), but this source hasn't been traced yet.

Les Amusemens. 1749. Prob. 216, p. 381 & fig. 97 on plate 8: Réduire les deux quarrés en un seul. Usual dissection of two adjacent squares, attributed to 'Sturmius', a German mathematician, i.e the previous entry.

Ozanam Montucla. 1778. Diverses démonstrations de la quarante-septieme du premier livre d'Euclide, ..., version 2. Fig. 27, plate 4. 1778: 288; 1803: 284; 1814: 241-243; 1840: 123-124. This is a version of the proof that (a + b)² = c² + 4(ab/2), but the diagram includes extra lines which produce the standard dissection of two adjacent triangles.

Crambrook. 1843. P. 4, no. 19: One Square to form two Squares -- ??

E. S. Loomis. The Pythagorean Proposition. 2nd ed., 1940; reprinted by NCTM, 1968. On pp. 194 195, he describes the usual dissection by two cuts as Geometric Proof 165 and gives examples back to 1849, Schlömilch.

Family Friend 2 (1850) 298 & 353. Practical Puzzle -- No. X. = Illustrated Boy's Own Treasury, 1860, Prob. 11, pp. 397 & 437. The larger square has twice the edge of the smaller and is shown divided into four, so this is clearly related to 6.AS.1, though the shape is considered as one piece, i.e. a P-pentomino, to be cut into three parts to make a square.

Magician's Own Book. 1857. To form a square, p. 261. = Book of 500 Puzzles, 1859, p. 75. An abbreviated version of Family Friend. Refers to dotted lines in the figure which are drawn solid.

Charades, Enigmas, and Riddles. 1860: prob. 31, pp. 60 & 65; 1862: prob. 32, pp. 136 & 142; 1865: prob. 576, pp. 108 & 155. Dissect a P-pentomino into three parts which make a square. Usual solution.

Peter Parley, the Younger. Amusements of Science. Peter Parley's Annual for 1866, pp. 139 155.

Pp. 143-144: "To form two squares of unequal size into one square, equal to both the original squares." Usual method, with five pieces. On pp. 146-148, he discusses the Theorem of Pythagoras and shows the dissection gives a proof of it.

P. 144: "To make two smaller squares out of one larger." Cuts the larger square along both diagonals and assembles the pieces into two squares.

Hanky Panky. 1872. To form a square, pp. 116-117. Very similar to Magician's Own Book.

Henry Perigal. Messenger of Mathematics 2 (1873) 104. ??NYS -- described in Loomis, op. cit. above, pp. 104-105 & 214, where some earlier possible occurrences are mentioned. He gives a dissection proof of the theorem of Pythagoras using the shapes that occur in the quadrisection of the square -- Section 6.AR. For sides a < b, perpendicular cuts through the centre are made in the square of side b so they meet the sides at distance (b-a)/2 from a corner. These pieces then fit around the square of side a to make a square of side c.

I invented a hinged version of this, in the 1980s?, which is described in: Greg N. Frederickson; Hinged Dissections: Swinging & Twisting; CUP, 2002, pp. 33-34. I am shown demonstrating this on Frederickson's website: www.cs.purdue.edu/homes/gnf/book2/Booknews2/singm.html .

I have seen the assembly of these four pieces and the square of edge a into the square on the hypotenuse in a photo of the Tomb of Ezekiel in the village of Al-Kifil, near Hillah, Iraq.

Mittenzwey. 1880.

Prob. 176, pp. 34 & 85; 1895?: 201, pp. 38 & 88; 1917: 201, pp. 35 & 84. Use the 10 pieces of 6.AS.1, as in Les Amusemens, to make squares of edge 1 and edge 2.

Prob. 180, pp. 34 & 86; 1895?: 205, pp. 39 & 89; 1917: 205, pp. 35-36 & 85. Cut a 2 x 2 and a 4 x 4 into five pieces which make a square. Both the problem and the solutions are inaccurately drawn. The smaller square has a 1, 2, 5 cut off, as for 6.AS.1. The larger square has the same cut off at the lower left and a 2, 4, 25 cut off at the lower right -- these two touch at the midpoint of the bottom edge -- leaving a quadrilateral with edges 4, 25, 5, 3 and two right angles. This is a variant of the standard five piece method.

Alf. A. Langley. Letter: Three-square puzzle. Knowledge 1 (9 Dec 1881) 116, item 97. Cuts two squares into five pieces which form a single square.

Alexander J. Ellis. Letter: The three-square puzzle. Knowledge 1 (23 Dec 1881) 166, item 146. Usual dissection of two adjacent squares, considered as one piece, into three parts by two cuts, which gives Langley's five pieces if the two squares are divided. Suppose the two squares are on a single piece of paper and are ABCD and DEFG, with E on side CD of the larger square ABCD. He notes that if one folds the paper so that B and F coincide, then the fold line meets the line ADG at the point H such that the desired cuts are BH and HF.

R. A. Proctor. Letter or editorial reply: Three square puzzle. Knowledge 1 (30 Dec 1881) 184, item 152. Says there have been many replies, cites Todhunter's Euclid, p. 266 and notes the pieces can be obtained by flipping the large square over and seeing how it cuts the two smaller ones.

R. A. Proctor. Our mathematical column: Notes on Euclid's first book. Knowledge 5 (2 May 1884) 318. "The following problem, forming a well-known "puzzle" exhibits an interesting proof of the 47th proposition." Gives the usual three piece form, as in Ellis.

B. Brodie. Letter: Superposition. Knowledge 5 (30 May 1884) 399, item 1273. Response to the above, giving the five piece version, as in Langley.

Hoffmann. 1893. Chap. III, no. 11: The two squares, pp. 93 & 125 126 = Hoffmann-Hordern, pp. 82-83, with photo. Smaller square has half the edge. The squares are viewed as a single piece. Photo on p. 83 shows The Five Squares Puzzle in paper with box, by Jaques & Son, 1870 1895, and an ivory version, with box, 1850-1900.

Loyd. Tit Bits 31 (3, 10 & 31 Oct 1896) 3, 25 & 75. General two cut version.

Herr Meyer. Puzzles. The Boy's Own Paper 19 (No. 937) (26 Dec 1896) 206 & (No. 948) (13 Mar 1897) 383. As in Hoffmann.

Benson. 1904. The two square puzzle, pp. 192 193.

Pearson. 1907. Part II, no. 108: Still a square, pp. 108 & 182. Smaller square has half the edge.

Loyd. Cyclopedia. 1914. Pythagoras' classical problem, pp. 101 & 352. c= SLAHP, pp. 15 16 & 88. The adjacent squares are viewed as one piece of wood to be cut. Uses two cuts, three pieces.

Williams. Home Entertainments. 1914. Square puzzle, p. 118. P-pentomino to be cut into three pieces to make a square. No solution given.

A. W. Siddons. Note 1020: Perigal's dissection for the Theorem of Pythagoras. MG 16 (No. 217) (Feb 1932) 44. Here he notes that the two cutting lines of Perigal's 1873 dissection do not have to go through the centre, but this gives dissections with more pieces. He shows examples with six and seven pieces. These cannot be hinged.

The Bile Beans Puzzle Book. 1933.

No. 37: No waste. Consider a square of side 2 extended by an isosceles triangle of hypotenuse 2. Convert to a square using two cuts.

No. 39: Square building. P-pentomino to square in two cuts.

Slocum. Compendium. Shows 4 - 5 piece square from Johnson Smith catalogue, 1935.

M. Adams. Puzzle Book. 1939. Prob. C.120: One table from two, pp. 154 & 185. 3 x 3 and 4 x 4 tiled squares to be made into a 5 x 5 but only cutting along the grid lines. Solves with each table cut into two pieces. (I think there are earlier examples of this -- I have just added this variant.)

Ripley's Puzzles and Games. 1966. Pp. 58-59.

Item 5. Two joined adjacent squares to a square, using two cuts and three pieces.

Item 6. Consider a 2 x 2 square with a 2, 2, 2 triangle attached to an edge. Two cuts and three pieces to make a square.

6.AS.2.a. TWO EQUAL SQUARES TO A SQUARE
Further subdivision of the pieces gives us 6.AS.4.
Jackson. Rational Amusement. 1821. Geometrical Puzzles, no. 13, pp. 25 & 85-86. Cut two equal squares each into two pieces to make a square.

Leske. Illustriertes Spielbuch für Mädchen. 1864? Prob. 172, p. 87. Cut one square into pieces to make two equal squares. Cuts along the diagonals.

Mittenzwey. 1880. Prob. 241, pp. 44 & 94; 1895?: 270, pp. 48 & 96; 1917: 270, pp. 44 & 92. As is Leske.
6.AS.3. THREE EQUAL SQUARES TO A SQUARE
Crambrook. 1843. P. 4, no. 21: One [Square to form] three [Squares] -- ??

"Student". Proposal [A pretty geometrical problem]. Knowledge 1 (13 Jan 1882) 229, item 184. Dissect an L-tromino into a square. Says there are 25 solutions -- editor says there are many more.

Editor. A pretty geometrical problem. Knowledge 1 (3 Mar 1882) 380. Says only the proposer has given a correct solution, which cuts off one square, then cuts the remaining double square into three parts, so the solution has four pieces. Says there are several other ways with four pieces and infinitely many with five pieces.

Hoffmann. 1893. Chap. III, no. 23: The dissected square, pp. 101 & 134 = Hoffmann Hordern, pp. 96-97, with photo. Cuts three squares identically into three pieces to form one square. Photo on p. 97 shows The Dissected Square, with box, by Jaques & Son, 1870-1895. Hordern Collection, p. 63, shows Arabian Puzzle, with box and some problem shapes to make, 1870 1890.

Loyd. Problem 3: The three squares puzzle. Tit Bits 31 (17 Oct, 7 & 14 Nov 1896) 39, 97 & 112. Quadrisect 3 x 1 rectangle to a square. Sphinx (i.e. Dudeney) notes it also can be done with three pieces.

M. Adams. Indoor Games. 1912. The divided square, p. 349 with figs. on pp. 346-347. 3 squares, 4 cuts, 7 pieces.

Loyd. Cyclopedia. 1914. Pp. 14 & 341. = SLAHP: Three in one, pp. 44 & 100. Viewed as a 3 x 1 rectangle, solution uses 2 cuts, 3 pieces. Viewed as 3 squares, there are 3 cuts, 6 pieces.

Johannes Lehmann. Kurzweil durch Mathe. Urania Verlag, Leipzig, 1980. No. 6, pp. 61 & 160. Claims the problem is posed by Abu'l-Wefa, late 10C, though other problems in this section are not strictly as posed by the historic figures cited. Two of the squares to be divided into 8 parts so all nine parts make a square. The solution has the general form of the quadrisection of the square of side 2 folded around to surround a square of side 1 (as in Perigal's(?) dissection proof of the Theorem of Pythagoras), thus forming the square of side 3. The four quadrisection pieces are cut into two triangles of sides: 1, 3/2, (1+2)/2 and 1, 3/2, (2-1)/2. Two of each shape assemble into a square of side 1 which can be viewed as having a diagonal cut and then cuts from the other corners to the diagonal, cutting off (2-1)/2 on the diagonal.

6.AS.3.a. THREE EQUAL 'SQUARES' TO A HEXAGON
Catel. Kunst-Cabinet. Vol. 2, 1793. Das Parallelogramm, pp. 14-15 & fig. 249 a,b,c,d on plate XII. This shows three squares, each dissected the same way into 4 pieces which will make a hexagon or two equal equilateral triangles. Consider a hexagon and connect three alternate vertices to the centre. Join up the same vertices and drop perpendiculars from the centre to three of the sides of the hexagon. However, close examination shows that the squares have dimensions 3/2 by 3. The figure of the three adjacent squares has the divisions between them hard to make out.

Bestelmeier. 1801. Item 292/293 -- Das Parallelogram. Almost identical to Catel, except the diagrams are reversed, and worse, several of the lines are missing. Mathematical part of the text is identical.

6.AS.4. EIGHT EQUAL SQUARES TO A SQUARE
Divide four of the squares in half diagonally.
Magician's Own Book. 1857. Prob. 8: The accommodating square, pp. 269 & 293. c= Landells, Boy's Own Toy-Maker, 1858, p. 144. = Book of 500 Puzzles, 1859, prob. 8, pp. 83 & 107. = Boy's Own Conjuring Book, 1860, prob. 7, pp. 230 & 256. = Illustrated Boy's Own Treasury, 1860, no. 24, pp. 399 & 439.

Hanky Panky. 1872. [Another square] of four squares and eight triangles, p. 120.

Cassell's. 1881. Pp. 92-93: The accommodating square. = Manson, 1911, p. 131.

Handy Book for Boys and Girls. Op. cit. in 6.F.3. 1892. Pp. 321-322: Square puzzle.

Hoffmann. 1893. Chap. III, no. 20: Eight squares in one, pp. 100 & 132 = Hoffmann Hordern, p. 94.

Wehman. New Book of 200 Puzzles. 1908. The accommodating square, p. 13. c= Magician's Own Book.

6.AS.5. RECTANGLE TO A SQUARE OR OTHER RECTANGLE
New section. See comment at 6.AS. The dissection of a 5 x 1 into five pieces which make a square is explicitly covered in 6.AS.1, and the other cases above can be viewed as dissections of 2 x 1, 3 x 1 and 8 x 1. There must be older examples of the general case??
Ozanam Montucla. 1778.

Avec cinq quarrés égaux, en former un seul. Prob. 18 & fig. 123, plate 15, 1778: 297; 1803: 292-293; 1814: 249-250; 1840: 127. 9 pieces. Remarks that any number of squares can be made into a square.

Prob. 19 & fig. 124-126, plate 15 & 16, 1778: 297-301; 1803: 293-296; 1814: 250 253; 1840: 127-129. Dissect a rectangle to a square.

Prob. 20 & fig. 125-126, plate 15 & 16, 1778: 301-302; 1803: 297; 1814: 253; 1840: 129. Dissect a square into 4, 5, 6, etc. parts which form a rectangle.

"Mogul". Proposal [A pretty geometrical problem]. Knowledge 1 (13 Jan 1882) 229, item 184. Dissect a rectangle into a square. Editor's comment in (3 Mar 1882) 380 says only the proposer has given a correct solution but it will be held over.

"Mogul". Mogul's Problem. Knowledge 1 (31 Mar 1882) 483. Gives a general construction, noting that if the ratio of length to width is  2, then it takes two cuts; if the ratio is in the interval (2, 5], it takes three cuts; if the ratio is in (5, 10], it takes four cuts; if the ratio is in (10, 17], it takes five cuts. In general if the ratio is in (n²+1, (n+1)²+1], it takes n+2 cuts.

Richard A. Proctor. Our puzzles; Knowledge 10 (Nov 1886) 9 & (Dec 1886) 39-40 & Solution of puzzles; Knowledge 10 (Jan 1887) 60-61. "Puzzle XII. Given a rectangular carpet of any shape and size to divide it with the fewest possible cuts so as to fit a rectangular floor of equal size but of any shape." He says this was previously given and solved by "Mogul". Solution notes that this is not the problem posed by "Mogul" and that the shape of the second rectangle is assumed as given. He distinguishes between the cases where the actual second rectangular area is given and where only its shape is given. Gives some solutions, remarking that more cuts may be needed if either rectangle is very long. Poses similar problems for a parallelogram.

Tom Tit, vol. 3. 1893. Rectangle changé en carré. en deux coups de ciseaux, pp. 175-176. = K, no. 24: By two cuts to change a rectangle into a square, pp. 64-65. Consider a square ABCD of side one. If you draw AA' at angle α to AB and then drop BE perpendicular to AA', the resulting three pieces make a rectangle of size sin α by csc α, where α must be  45⁰, so the rectangle cannot be more than twice as long as it is wide. If one starts with such a rectangle ABCD, where AB is the length, then one draws AA' so that DA' is the geometric mean of AB and AB - AD. Dropping CE perpendicular to AA' gives the second cut.

Dudeney. Perplexities column, no. 109: A cutting-out puzzle. Strand Magazine 45 (No. 265) (Jan 1913) 113 & (No. 266) (Feb 1913) 238. c= AM, prob. 153 -- A cutting-out puzzle, pp. 37 & 172. Cut a 5 x 1 into four pieces to make a square. AM states the generalized form: if length/breadth is in [(n+1)², n²), then it can be done with n+2 pieces, of which n-1 are rectangles of the same breadth but having the desired length. The cases 1 x (n+1)² are exceptional in that one of pieces vanishes, so only n+1 pieces are needed. He doesn't describe this fully and I think one can change the interval above to ((n+1)², n²].

Anonymous. Two dissection problems, no. 1. Eureka 13 (Oct 1950) 6 & 14 (Oct 1951) 23. An n-step is formed by n lines of unit squares of lengths 1, 2, ..., n, with all lines aligned at one end. Hence a 1-step is a unit square, a 2-step is an L-tromino and an n step is what is left when an (n-1)-step is removed from a corner of an n x n square. Show any n-step can be cut into four pieces to make a square, with three pieces in one case. Cut parallel to a long side at distance (n+1)/2 from it. The small piece can be rotated 180^o about a corner to make an n x (n+1)/2 rectangle. Dudeney's method cuts this into three pieces which make a square, and the cuts do not cut the small part, so we can do this with a total of four pieces. When n = 8, the rectangles is 8 x 9/2, which is similar to 16 x 9 which can be cut into two pieces by a staircase cut, so the problem can be done with a total of three pieces. A little calculation shows this is the only case where n x (n+1)/2 is similar to k² x (k-1)².

Lorraine Mottershead. Investigations in Mathematics. Blackwell, Oxford, 1985. P. 105. Dissect a 2 x 5 rectangle into four pieces that make a square.
6.AT. POLYHEDRA AND TESSELLATIONS
These have been extensively studied, so I give only the major works. See 6.AA for nets of polyhedra.
6.AT.1. REGULAR POLYHEDRA
Gwen White. Antique Toys and Their Background. Batsford, 1971. (Reprinted by Chancellor Press, London, nd [c1989].) P. 9 has a sketch of "Ball of stone, Scotland", which seems to be tetrahedral and she says: "... one of the earliest toys known is a stone ball. Perhaps it is not a plaything, no one knows why it was made, but it is a convenient size to hold in the hand."

Dorothy N. Marshall. Carved stone balls. Proc. Soc. of Antiquaries of Scotland 108 (1976-7) 40-72. Survey of the Scottish neolithic carved stone balls. Lists 387 examples in 36 museums and private collections, mostly of 70mm diameter and mostly from eastern Scotland. Unfortunately Marshall is not interested in the geometry and doesn't clearly describe the patterns -- she describes balls with 3, 4, 5, 6, 7, 8, 9, 10 - 55 and 70 - 160 knobs, but emphasises the decorative styles. From the figures, there are clearly tetrahedral, cubical, dodecahedral(?) and cubo-octahedral shapes. Many are in the National Museum of Antiquities of Scotland (= Royal Museum, see below), but the catalogue uses a number of unexplained abbreviations of collections.

Royal Museum of Scotland, Queen Street, Edinburgh, has several dozen balls on display, showing cubical, tetrahedral, octahedral and dodecahedral symmetry, and one in the form of the dual of the pentagonal prism. [This museum has now moved to a new building beside its other site in Chambers Street and has been renamed the Museum of Scotland. When I visited in 1999, I was dismayed to find that only three of the carved stone balls were on display, in a dimly lit case and some distance behind the glass so that it was difficult to see them. Admittedly, the most famous example, the tetrahedral example with elaborate celtic decorative spirals, NMA AS10 from Glasshill, Towie, Aberdeenshire, is on display -- photo in [Jenni Calder; Museum of Scotland; NMS Publishing, 1998, p. 21]. They are on Level 0 in the section called In Touch with the Gods.]

Ashmolean Museum, Oxford, has six balls on display in case 13a of the John Evans Room. One is tetrahedral, three are cubical, one is dodecahedral and one is unclear.

Keith Critchlow. Time Stands Still -- New Light on Megalithic Science. Gordon Fraser, London, 1979. Chap. 7: Platonic spheres -- a millennium before Plato, pp. 131 149. He discusses and depicts Neolithic Scottish stones carved into rounded polyhedral shapes. All the regular polyhedra and the cubo octahedron occur. He is a bit vague on locations -- a map shows about 50 discovery sites and he indicates that some of these stones are in the Ashmolean Museum, Dundee City Museum and 'in Edinburgh'. Likewise, the dating is not clear -- he only says 'Neolithic' -- and there seem to be no references.

D. V. Clark. Symbols of Power at the Time of Stonehenge. National Museum of Antiquities, Edinburgh, 1985. Pp. 56-62 & 171. ??NYS -- cited by the Christie's Catalogue, below.

Robert Dixon. Mathographics. Blackwell, 1987, fig. 5.1B, p. 130, is a good photo of the Towie example.

Anna Ritchie. Scotland BC. HMSO, Edinburgh, for Scottish Development Department -- Historic Buildings and Monuments, 1988.

P. 8 has a colour photo of a neolithic cubical ball from the Dark Age fort of Dunadd, Argyll.

P. 14 has a colour photo of a cubical and a tetrahedral ball from Skara Brae, Orkney Islands, c-2800.

Simant Bostock of Glastonbury has made a facsimile of the Towie example, casts of which are available from Glastonbury Film Office, 3 Market Place, Glastonbury, Somerset, BA6 9HD; tel: [phone]. You can also contact him at 24 Northload Street, Glastonbury, Somerset, BA6 9JJ; tel: [phone] and he has a mail order catalogue. Since he worked from photographs, there are some slight differences from the original, and the facsimile is slightly larger.

Three examples of tetrahedral stone balls were in Christie's South Kensington antiquities sale of 12 Apr 2000, lots 124 and 125 (2 balls), p. 62, with colour photo of item 124 and the better example in lot 124 on p. 63. (Thanks to Christine Insley Green for a copy of the catalogue.) The descriptive text says 'their exact use is unclear'. Cites Clark, above.

The British Museum has icosahedral dice from Egypt, dated -700/-500.

Moritz Cantor. Vorlesungen über Geschichte der Mathematik. Vol. I, 4th ed., 1906, pp. 174 176. He feels all the regular solids were known to Pythagoras, with the tetrahedron, cube and octahedron having been known long before. Says to see various notices by Count Leopold Hugo in CR 77 for a bronze dodecahedron, a work by Conze on a Celtic bronze example and the paper of Lindemann, below, for a north Italian example. However, he says the dates of these are not determined and I think these are now all dated to later Roman times -- see below. He also notes that moderately regular dodecahedra and icosahedra occur in mineral deposits on Elba and in the Alps and wonders if Pythagoras could have known of these.

Thomas L. Heath. Note about Scholium 1 of Book XIII of Euclid. The Thirteen Books of Euclid's Elements; trans. & ed. by Thomas L. Heath; (1908?); 2nd ed., (1926); Dover, 3 vols., 1956, vol. 3, p. 438. "And it appears that dodecahedra have been found, of bronze or other material, which may belong to periods earlier than Pythagoras' time by some centuries (for references see Cantor's Geschichte der Mathematik I₃, pp. 175-6)."

HGM I 160 cites Hugo and Lindemann, dating the Monte Loffa example as -1000/-500.

HGM I 162 discusses the Scholium, giving it as: "the five so-called Platonic figures, which, however, do not belong to Plato, three of the five being due to the Pythagoreans, namely the cube, the pyramid, and the dodecahedron, while the octahedron and icosahedron are due to Theaetetus". He cites Heiberg's Euclid, vol. v., p. 654.

Thomas, SIHGM I 223 says "A number of objects of dodecahedral form have survived from pre-Pythagorean days." But he gives no details or references. Cf Heath's note to Euclid, above.

Plato ( 427/ 347). Timaeus. c-350. Page references are based on the 1578 edition of Plato which has been used for all later references: pp. 54-56. I use the version in: Edith Hamilton and Huntington Cairns, eds; The Collected Dialogues of Plato including the Letters; Bollingen Series LXXI, Pantheon Books (Random House), (1961), corrected 2nd ptg, 1963, which is the translation of Benjamin Jowett in his The Dialogues of Plato, OUP, (1872), 4th revised ed, 1953. Discusses the regular polyhedra, describing the construction of the tetrahedron, octahedron, icosahedron and cube from triangles since he views the equilateral triangle as made from six 30-60-90 triangles and the square made from eight 45-45-90 triangles. "There was yet a fifth combination which God used in the delineation of the universe with figures of animals." He then relates the first four to the elements: tetrahedron -- fire; octahedron -- air; icosahedron -- water; cube -- earth. [These associations are believed to derive from the Pythagoreans.] However, these associations contribute very little to the rest of the dialogue. The incidental appearance of the dodecahedron lends support to the belief that it was discovered or became known after the initial relation between regular polyhedra and the elements had been established and had to be added in some ad hoc manner. [It is believed that the later Pythagoreans related it to the universe as a whole.]

Scholium 1 of Book XIII of Euclid. Discussed in: The Thirteen Books of Euclid's Elements; op. cit. above, vol. 3, p. 438. Heath's discussion of the Scholia in vol. 1, pp. 64-74, indicates this may be c600. The Scholium asserts that only the tetrahedron, cube and dodecahedron were known to the Pythagoreans and that the other two were due to Theaetetus. Heath thinks the Pythagoreans had all five solids (cf his note to IV.10, vol. 2, pp. 97-100) and the Scholium is taken from Geminus, who may have been influenced by the fact that Theaetetus was the first to write about all five solids and hence the first to write much about the latter two polyhedra.

Stefano de'Stefani. Intorno un dodecaedro quasi regolare di pietra a facce pentagonali scolpite con cifre scoperto nelle antichissime capanne di pietra del Monte Loffa. Atti del Reale Istituto Veneto di Scienze e Lettere, (Ser. 6) 4 (1885) 1437-1459 + plate 18. Separately reprinted by G. Antonelli, Venezia, 1886, which has pp. 1-25 and Tavola 18. This describes perhaps the oldest known reasonably regular dodecahedron, in the Museo Civico di Storia Naturale, Palazzo Pompei, Largo Porta Vittoria 9, Verona, Italy, in the central case of Sala XIX. This is discussed by Herz-Fischler [op. cit. below, p. 61], I have been to see it and the Director, Dr. Alessandra Aspes, has kindly sent me a slide and a photocopy of this article.

The dodecahedron was discovered in 1886 at Monte Loffa, NE of Verona, and has been dated as far back as -10C, but is currently considered to be -3C or -2C [Herz Fischler, p. 61]. Dr. Aspes said the site was inhabited by tribes who had retreated into the mountains when the Romans came to the area, c-3C. These tribes were friendly with the Romans and were assimilated over a few centuries, so it is not possible to know if this object belongs to the pre-Roman culture or was due to Roman influence. She dates it as  4C/ 1C. The stone apparently was cut with a bronze saw and these existed before the Roman incursion (stated in Lindemann, below). It is clearly not perfectly regular -- some of the face angles appear to be 90^o and some edges are clearly much shorter than others. But it also seems clear that it is an attempt at a regular dodecahedron -- the faces are quite flat. Its faces are marked with holes and lines, but their meaning and the function of the object are unknown. de'Stefani conjectures it is a kind of die. Lindemann notes that the symbols are not Etruscan nor Greek, but eventually gets to an interpretation of them, which seems not too fanciful, using the values: 3, 6, 9, 10, 12, 15, 16, 20, 21, 24, 60, 300. (Are there any ancient Greek models of the regular polyhedra?) But see also the carved stone balls above.

About 90 examples of a Roman dodecahedron have been found at Roman sites, north of the Alps, from Britain to the Balkans, dating about 200-400. These are bronze and hollow, but also each face has a hole in it, almost always circular, and each corner has a knob at it, making it look like it could be used for Hamilton's Icosian game! The shape is quite precise.

The Society of Antiquaries, London, has the largest extant specimen, dug up on the north side of the Church of St. Mary, Carmarthen, in 1768 and presented to the Society about 1780. The edge length is 2 1/5 in (= 56mm) and, unusually, has plain faces -- almost all examples have some incised decoration on the faces. [Rupert Bruce-Mitford; The Society of Antiquaries of London Notes on Its History and Possessions; The Society, 1951, p. 75, with photograph as pl. XXIV (b) on p. 74.

The Gallo-Romeins Museum (Kielenstraat 15, B-3700 Tongeren, Belgium; Tel: [phone]) has an example which is the subject of an exhibition and they have produced facsimiles for sale. "The precise significance and exact use of this object have never been explained and remain a great mystery." Luc de Smet says the bronze facsimile is slightly smaller than the original and that the Museum also sells a tin and a bronzed version of the original size.

Other examples are in the Newcastle University Museum (only about half present) and the Hunt Museum, Limerick.

C. W. Ceram. Gods, Graves and Scholars. Knopf, New York, 1956, pp. 26-29. 2nd ed., Gollancz, London, 1971, pp. 24-25. In the first edition, he illustrated this as an example of the mysterious objects which archaeologists turn up and said that it had been described as a toy, a die, a model for teaching measurement of cylinders, a candleholder. His picture shows one opening as being like a key-hole. In the second edition, he added that he had over a hundred suggestions as to what it was for and thinks the most probable answer is that it was a musical instrument.

Jacques Haubrich has recently sent an example of a hollow cubical stone object with different size holes in the faces, apparently currently made in India, sold as a candleholder.

See Thomas, SIHGM I 216 225, for brief references by Philolaus, Aëtius, Plato, Iamblichus.

Euclid. Elements. c-300. Book 13, props. 13 18 and following text. (The Thirteen Books of Euclid's Elements, edited by Sir Thomas L. Heath. 2nd ed., (CUP, 1925??), Dover, vol. 3, pp. 467 511.) Constructs the 5 regular polyhedra in a sphere, compares them. In Prop. 18, he continues "I say next that no other figure, besides the said five figures, can be constructed which is contained by equilateral and equiangular figures equal to one another."

Leonardo Pisano, called Fibonacci (c1170->1240). La Practica Geometriae. 1221. As: La Practica Geometriae di Leonardo Pisano secondo la lezione del Codice Urbinate n^o. 202 della Biblioteca Vaticana. In: Scritti di Leonardo Pisano; vol. II, ed. and pub. by B. Boncompagni; Tipografia delle Scienze Matematiche e Fisiche, Rome, 1862, pp. 1 224. On p. 159, he says there are many polyhedra and mentions there are ones with 8, 12 and 20 faces which Euclid constructs in a sphere in his book XIIII. On pp. 161-162, he describes division in mean and extreme ratio and the construction of the regular pentagon in a circle, then says you can construct, in a sphere, a solid with 20 equilateral triangular faces or with 12 pentagonal faces. After some discussion, he says you can also construct solids with 4, 6, 8, 12, 20 faces, in a sphere. Division in mean and extreme ratio and the construction of the icosahedron are later covered in detail on pp. 196-202. His only drawings of solids are of cubes and pyramids.

Drawings of all the regular polyhedra are included in works, cited in 6.AA, 6.AT.2 and 6.AT.3, by della Francesca (c1480 & c1487), Pacioli (1494), Pacioli & da Vinci (1498), Dürer (1525), Jamnitzer (1568), and Kepler (1619).

F. Lindemann. Zur Geschichte der Polyeder und der Zahlzeichen. Sitzungsber. der math. phys. Classe k. b. Akademie der Wissenschaften zu München 26 (1896) 625 758 & plates I-IX. Discusses and illustrates many ancient polyhedra. Unfortunately, most of these are undated and/or without provenance. He generally dates them as -7C/5C.

A bronze rhombic triacontahedron, which he dates as first half of the first millennium AD.

Roman knobbed dodecahedra, which he describes as Celtic, going back to the La Tène period (Bronze Age) -- these are now dated to late Roman times. He lists 26 examples listed from the works of Conze and Hugo (cf Cantor, above).

A dodecahedral die; an irregular rhombi-cubo-octahedral die; a bronze dodecahedral die (having two 1s, three 2s, two 3s, one 4 and four 5s).

The Verona dodecahedron (from de'Stefani), which he dates as -1000/-500.

An enamelled icosahedron in Turin with Greek letters on the faces.

An octagonal bipyramid (elongated) from Meclo, South Tyrol, marked with a form of Roman numerals in a somewhat irregular order. It is dated to before the Barbarian migrations.

Three bronze cubo-octahedra.

He then does a long analysis of north-Italian culture and its relations to other cultures and of their number symbols, eventually obtaining an interpretation of the symbols on the Monte Loffa dodecahedron, which he then justifies further with Pythagorean number relations.

Roger Herz-Fischler. A Mathematical History of Division in Extreme and Mean Ratio. Wilfrid Laurier University Press, Waterloo, Ontario, 1987. Corrected and extended as: A Mathematical History of the Golden Number; Dover, 1998. P. 61 discusses the history of the dodecahedron and refers to the best articles on the history of polyhedra. Discusses the Verona dodecahedron, see above.

Judith V. Field. Kepler's Geometrical Cosmology. Athlone Press, London, 1988. This gives a good survey of the work of Kepler and his predecessors. In particular, Appendix 4: Kepler and the rhombic solids, pp. 201-219, is most informative. Kepler described most of his ideas several times and this book describes all of them and the relationships among the various versions.

The regular polyhedra in four dimensions were described by Ludwig Schläfli, c1850, but this was not recognised and in the 1880s, several authors rediscovered them.

H. S. M. Coxeter. Regular skew polyhedra. Proc. London Math. Soc. (2) 43 (1937) 33-62. ??NYS -- cited and discussed by Gott, qv.

J. R. Gott III. Pseudopolyhedrons. AMM 74:5 (May 1967) 497-504. Regular polyhedra have their sum of face angles at a vertex being less than 360^o and approximate to surfaces of constant positive curvature, while tessellations, with angle sum equal to 360^o, correspond to surfaces of zero curvature. The pseudopolyhedra have angle sum greater than 360^o and approximate to surfaces of negative curvature. There are seven regular pseudopolyhedra. Each is a periodic structure. He subsequently discovered that J. F. Petrie and Coxeter had discovered three of these in 1926 and had shown that they were the only examples satisfying an additional condition that arrangement of polygons at any vertex have rotational symmetry, and hence that the dihedral angles between adjacent faces are all equal. Coxeter later refers to these structures as regular honeycombs. Some of Gott's examples have some dihedral angles of 180^o. Two of these consist of two planes, with a regular replacement of pieces in the planes by pieces joining the two planes. The other five examples go to infinity in all directions and divide space into two congruent parts. He makes some remarks about extending this to general and Archimedean pseudopolyhedra.

6.AT.2 STAR AND STELLATED POLYHEDRA
I have heard it stated that Kircher was the first to draw star polygons.

Paolo Uccello (1397 1475). Mosaic square on the floor at the door of San Pietro in San Marco, Venice. 1425 1430. (This doorway is not labelled on the maps that I have seen -- it is the inner doorway corresponding to the outer doorway second from the left, i.e. between Porta di Sant'Alipio and Porta di San Clemente, which are often labelled.) This seems to show the small stellated dodecahedron {5/2, 5}. This mosaic has only recently (1955 & 1957) been attributed to Uccello, so it can only be found in more recent books on him. See, e.g., Ennio Flaiano & Lucia Tongiorgi Tomase; L'Opera Completa di Paolo Uccello; Rizzoli, Milan, 1971 (and several translations). The mosaic is item 5.A: Rombo con elementi geometrici in the Catalogo delle Opere, with description and a small B&W picture on p. 85. [Bokowski & Wills, below, give the date 1420.]

Coxeter [Elem. der Math. 44 (1989) 25 36] says it "is evidently intended to be a picture of this star polyhedron."

However, J. V. Field tells me that the shape is not truly the small stellated dodecahedron, but just a 'spiky' dodecahedron. She has examined the mosaic and the 'lines' of the pentagrams are not straight. [The above cited photo is too small to confirm this.] She says it appears to be a direct copy of a drawing in Daniele Barbaro; La Practica della Perspettiva; Venice, 1568, 1569, see below, and is most unlikely to be by Uccello. See Field, Appendix 4, for a discussion of early stellations.

In 1998, I examined the mosaic and my photos of it and decided that the 'lines' are pretty straight, to the degree of error that a mason could work, and some are dead straight, so I agree with Coxeter that it is intended to be the small stellated dodecahedron. I now have a postcard of this. However, I have recently seen a poster of a different mosaic of the same shape which is distinctly irregular, so the different opinions may be based on seeing different mosaics!

Both mosaics are viewed directly onto a pentagonal pyramid, but the pyramids are distinctly too short in the poster version. The only spiky dodecahedron in Barbaro is on p. 111, fig. 52, and this is viewed looking at a common edge of two of the pyramids and the pyramids are distinctly too tall, so this is unlikely to be the source of the mosaics. The 'elevated dodecahedron' in Pacioli & da Vinci, plate XXXI, f. CVI-v, has short pyramids and looks quite like the second mosaic, but it is viewed slightly at an angle so the image does not have rotational symmetry. If anyone is in Venice, perhaps they could check whether there are two (or more?) mosaics and get pictures of them.

Luca Pacioli & Leonardo da Vinci. Untitled MS of 1498, beginning: Tavola dela presente opera e utilissimo compendio detto dela divina proportione dele mathematici discipline e lecto -- generally called De divina proportione. Ill. by Leonardo da Vinci. See the entry in 6.AT.3 for fuller details of the facsimiles and details about which plates are in which of the editions.

Discussed by Mackinnon (see in 6.AT.3 below) and Field, pp. 214-215. Clearly shows the stella octangula in one of the superb illustrations of Leonardo, described as a raised or elevated octahedron (plates XVIIII & XX). Field, p. 214, gives the illustration. None of the other raised shapes is a star, but the raised icosahedron is close to a star shape.

Barbaro, Daniele (1514-1570). La Practica della Perspettiva. Camillo & Rutilio Borgominieri, Venice, 1568, 2nd ptg, 1569. (Facsimile from a 1569 copy, Arnaldo Forni, Milan, 1980. The facsimile's TP doesn't have the publication details, but they are given in the colophon. Various catalogues say there are several versions with dates on the TP and colophon varying independently between 1568 and 1569. A version has both dates being 1568, so this is presumed to be the first appearance. Another version has an undated title in an elaborate border and this facsimile must be from that version.) P. 111 has a dodecahedron with pyramids on each face, close to, but clearly not the stellated dodecahedron. P. 112 has an icosahedron with pyramids on each face, again close to, but clearly not the stellated icosahedron. I would have expected a reasonably accurate drawing, but in both drawings, several of the triples of segments which should lie on a single straight line clearly do not. P. 113 shows an icosi-dodecahedron with pyramids on the triangular faces. If the pyramids extended the edges of the pentagons, this would produce the dodecahedron! But here the pyramids distinctly point much further out and the overall perspective seems wrong. [Honeyman, no. 207, observing that some blocks come from the 1566 edition of Serlio which was dedicated to Barbaro.]

Wentzel Jamnitzer (or Jamitzer). Perspectiva Corporum Regularum. With 50 copper plates by Jost Amman. (Nürnberg, 1568.) Facsimile by Akademische Druck- u. Verlagsanstalt, Graz, 1973. [Facsimiles or reprints have also been issued by Alain Brieux, Paris, 1964 and Verlag Biermann und Boukes, Frankfurt, 1972.]

This includes 164 drawings of polyhedra in various elaborations, ranging from the 5 regular solids through various stellations and truncations, various skeletal versions, pseudo-spherical shapes and even rings. Some polyhedra are shown in different views on different pages. Nameable objects, sometimes part of larger drawings, include: tetrahedron, cubo-octahedron, truncated tetrahedron, stella octangula, octahedron, cube, truncated octahedron, rhombi-cuboctahedron, compound of a cube and an octahedron (not quite correct), great rhombi-cuboctahedron, icosahedron, great dodecahedron, dodecahedron, icosi-dodecahedron, rhombi-icosi-dodecahedron, truncated cube, and skeletal versions of: stella octangula, octahedron, cube, icosahedron, dodecahedron, icosi-dodecahedron. There are probably some uniform polyhedra, but I haven't tried to identify them, and some of the truncated and stellated objects might be nameable with some effort.

J. Kepler. Letter to Herwart von Hohenberg. 6 Aug 1599. In: Johannes Kepler Gesammelte Werke, ed. by M. Caspar, Beck, Munich, 1938. Vol 14, p. 21, letter 130, line 457. ??NYS. Cited by Field, op. cit. below. Refers to (small??) stellated dodecahedron.

J. Kepler. Letter to Maestlin (= Mästlin). 29 Aug 1599. Ibid. Vol. 14, p. 43, letter 132, lines 142-145. ??NYS. Cited by Field, below, and in [Kepler's Geometrical Cosmology; Athlone Press, London, 1988, p. 202]. Refers to (small??) stellated dodecahedron.

J. Kepler. Harmonices Mundi. Godfrey Tampach, Linz, Austria, 1619; facsimile: (Editions) Culture et Civilization, Brussels, 1968 (but my copy is missing three plates!) [Editions probably should have É, but my only text which uses the word Editions is a leaflet in English.] = Joannis Kepleri Astronomi Opera Omnia; ed. Ch. Frisch, Heyder & Zimmer, Frankfurt & Erlangen, 1864, vol. 5. = Johannes Kepler Gesammelte Werke; ed. by M. Caspar, Beck, Munich, 1938, vol. 6, ??NYS. Book II. Translated by J. V. Field; Kepler's star polyhedra; Vistas in Astronomy 23 (1979) 109 141.

Prop. XXVI, p. 60 & figs. Ss & Tt on p. 53. Describes both stellated dodecahedra, {5/2, 5} and {5/2, 3}. This is often cited as the source of the stella octangula, but the translation is referring to an 'eared cube' with six octagram faces and the stella octangula is clearly shown by Pacioli & da Vinci and by Jamnitzer.

Louis Poinsot. Mémoire sur les polygones et les polyèdres. J. de L'École Polytechnique 4 (1810) 16 48 & plate opp. p. 48. Art. 33 40, pp. 39 42, describe all the regular star polyhedra. He doesn't mention Kepler here, but does a few pages later when discussing Archimedean polyhedra.

A. L. Cauchy. Recherches sur les polyèdres. J. de L'École Polytechnique 16 (1813) 68 86. ??NYS. Shows there are no more regular star polyhedra and this also shows there are no more stellations of the dodecahedron.

H. S. M. Coxeter, P. Du Val, H. T. Flather & J. F. Petrie. The Fifty-Nine Icosahedra. Univ. of Toronto Press, 1938; with new Preface by Du Val, Springer, 1982. Shows that there are just 59 stellations of the icosahedron. They cite earlier workers: M. Brückner (1900) found 12; A. H. Wheeler (1924) found 22.

Dorman Luke. Stellations of the rhombic dodecahedron. MG 41 (No. 337) (Oct 1957) 189 194. With a note by H. M. Cundy which says that the first stellation is well known (see 6.W.4) and that the second and third are in Brückner's Vielecke und Vielfläche, but that the new combinations shown here complete the stellations in the sense of Coxeter et al.

J. D. Ede. Rhombic triacontahedra. MG 42 (No. 340) (May 1958) 98 100. Discusses Coxeter et al. and says the main process generates 8 solids for the icosahedron. He finds that the main process gives 13 for the rhombic triacontahedron, but makes no attempt to find the analogues of Coxeter et al.'s 59.

6.AT.3. ARCHIMEDEAN POLYHEDRA
Archimedes discovered the Archimedean solids, -3C, but his work is lost. Heron quotes some of it and Pappus summarises it. See HGM II 98-101.

Hero of Alexandria (c150). Definitiones. IN: Heronis Alexandrini Opera quae supersunt omnia; Vol. IV, Heronis Definitiones Cum Variis Collectionibus Heronis Quae Feruntur Geometrica, ed. by J. L. Heiberg, Teubner, 1912, pp. 64-67. Heath, HGM I 294-295 has a translation, but it doesn't give the complete text which seems open to two interpretations. The German goes: Archimedes aber sagt, es gebe in ganzen dreizehn Körper, die in einer Kugel eingeschreiben werden können, indem er ausser den genannten fünf noch acht hinzufügt; von diesen habe auch Platon das Tessareskaidekaeder gekannt, dies aber sei ein zweifaches, das eine aus acht Dreiecken und sechs Quadraten zusammengesetzt, aus Erde und Luft, welches auch einige von den Alten gekannt hätten, das andere umgekehrt aus acht Quadraten und sechs Dreiecken, welches schwieriger zu sein scheint. My translation: But Archimedes said, there are in total 13 bodies, which can be inscribed in a sphere, as he added eight beyond the named five [regular solids, which he had just defined]; but of these Plato knew the 14-hedron, however this is a double, one is composed of eight triangles and six squares, from Earth and Air, which some of the ancient also knew, the other conversely [is composed] of eight squares and six triangles, which seems to be more difficult.

Note that Hero has got the numbers wrong - Archimedes found 13 more than the 5 regular solids. Secondly, the 'more difficult' solid does not exist! Heath notes this and suggests that either the truncated cube or the truncated octahedron was intended. The question of interpretation arises at the first semicolon -- is this continuing the statement of Archimedes or is Hero commenting on Archimedes' results? Heath seems to say Archimedes is making the attribution to Plato, but see below. MacKinnon, below, seems to be say this is being made by Hero. Heath's discussion on HGM II 100 says "We have seen that, according to Heron, two of the semi-regular [i.e. Archimedean] solids had already been discovered by Plato" undoubtedly using the method of truncation. However, I don't see that Heron is saying that Plato discovered the cubo-octahedron and the other solid, only that he knew it. Mackinnon says "Plato is said by Heron to have discovered the cuboctahedron by making a model of it from a net." But I don't see that Heron says this.

Pappus. Collection. c290. Vol. 19. In: SIHGM II 194 199. Describes the 13 Archimedean solids. "..., but also the solids, thirteen in number, which were discovered by Archimedes and are contained by equilateral and equiangular, but not similar, polygons." He then describes each one. Pappus' work has survived in a single MS (Vat. gr. 218) of the 10C in the Vatican and was not copied until 1550, but see Mackinnon, pp. 175-177, on whether it had been seen by Piero. For the history of this MS, see also: Noel M. Swerdlow; The recovery of the exact sciences of antiquity: mathematics, astronomy, geography; IN: Anthony Grafton, ed.; Rome Reborn The Vatican Library and Renaissance Culture Catalog of an exhibition at the Library of Congress, Washington, D.C., Jan. 6   Apr. 30, 1993; Library of Congress, Washington & Yale University Press, New Haven & London; in association with the Biblioteca Apostolica Vaticana; 1993; pp. 137-139. [This exhibition is on-line at www.ibiblio.org/expo/vatican.exhibit/vatican.exhibit.html.]

R. Ripley. Believe It Or Not. 18th series, Pocket Books, NY, 1971. P. 116 asserts the Romans used dice in the shape of cubo octahedra.

The British Museum, Room 72, Case 9, has two Roman cubo-octahedral dice on display.

F. Lindemann, op. cit. in 6.AT.1, 1896, describes and illustrates an antique rhombic triacontahedron, possibly a die, possibly from the middle of the Byzantine era.

Nick Mackinnon. The portrait of Fra Luca Pacioli. MG 77 (No. 479) (Jul 1993) plates 1-4 & pp. 129-219. Discusses the various early authors, but has mistakes.

della Francesca. Trattato. c1480. Ff. 105r - 117v (224-250) treats solid bodies, discussing all the regular polyhedra, with figures, though Arrighi gives only a projection of the octahedron. Discusses and gives good diagrams of the truncated tetrahedron and cubo octahedron, apparently the first drawings of any Archimedean polyhedra. Jayawardene refers to the cubo-octahedron as a truncated cube.

Davis notes that Pacioli's Summa, Part II, ff. 68v - 73v, prob. 1-56, are essentially identical to della Francesca's Trattato, ff. 105r - 120r.

Piero della Francesca. Libellus de Quinque Corporibus Regularibus. c1487 [Davis, p. 44, dates it to 1482-1492]. Piero would have written this in Italian and it is believed to have been translated into Latin by Matteo da Borgo [Davis, p. 54], who improved the style. First post-classical discussion of the Archimedean polyhedra, but it was not published until an Italian translation (probably by Pacioli) was printed in Pacioli & da Vinci, qv, in 1509, as: Libellus in tres partiales tractatus divisus quae corpori regularium e depēdentiū actine perscrutatiōis ..., ff. 1 27. A Latin version was discovered by J. Dennistoun, c1850, and rediscovered by Max Jordan, 1880, in the Urbino manuscripts in the Vatican -- MS Vat.Urb.lat. 632; the Duke of Urbino was a patron of Piero and in the MS, Piero asks that it be placed by his De Prospectiva Pingendi in the Duke's library. This was published by Girolamo Mancini in: L'opera "De corporibus regularibus" di Pietro Franceschi detto della Francesca usurpata da Fra Luca Pacioli, Memorie della Reale Accademia dei Lincei, Classe di Scienze Morali, Storiche e Filologiche (5) 14:7B (1915) 441-580 & 8 plates, also separately published by Tipografia della Reale Accademia dei Lincei, Rome, 1916. Davis identifies 139 problems in this, of which 85 (= 61%) are taken from the Trattato. There is debate as to how much of this work is due to Piero and how much to Pacioli. The Latin text differs a bit from the Italian. See the works of Taylor and Davis in Section 1 under Pacioli and the discussion on della Francesca's Trattato and Pacioli's Summa in the common references.

He describes a sphere divided into 6 zones and 12 sectors. Mackinnon says Piero describes seven of the Archimedean polyhedra, but without pictures, namely: cuboctahedron, truncated tetrahedron, truncated cube, truncated octahedron, truncated dodecahedron, truncated icosahedron, rhombi-cuboctahedron. Field, op. cit. in 6.AT.1, p. 107, says Piero gives six of the Archimedean polyhedra. In recent lectures Field has given a table showing which Archimedean polyhedra appear in Piero, Pacioli, Dürer and Barbaro and this lists just the first six of the above as being in Piero. I find just the five truncated regular polyhedra -- see above for the cubo-octahedron -- and there is an excellent picture of the truncated tetrahedron on f. 22v of the printed version. Mancini gives different diagrams than in the 1509 printed version, including clear pictures of the truncated icosahedron and the truncated dodecahedron. della Francesca clearly has the general idea of truncation. An internet biographical piece, apparently by, or taken from, J. V. Field, (http://www history.mcs.st andrews.ac.uk/history/Mathematicians.Francesca.html), shows that the counting is confused by the presence of the cubo-octahedron in the Trattato but not in the Libellus. So della Francesca rediscovered six Archimedean polyhedra, but only five appear in the Libellus. The work of Pappus was not known at this time.

Pacioli. Summa. 1494.

f. 4. Brief descriptions of the cubo-octahedron, truncated tetrahedron, icosidodecahedron, truncated icosahedron. No drawings.

Part II, ff. 68v - 72r, sections 2 (unlabelled) - 35. Discussion and some crude drawings of the regular polyhedra, the truncated tetrahedron and the cubo-octahedron. Mackinnon says these are the first printed illustrations of any Archimedean polyhedra. Davis notes that Part II, ff. 68v - 73v, prob. 1-56, are essentially identical to della Francesca's Trattato, ff. 105r - 120r.

Jacopo de'Barbari or Leonardo da Vinci. Portrait of Fra Luca Pacioli. 1495. In the Museo Nazionale di Capodimonte, Naples. The upper left shows a glass rhombi-cuboctahedron half filled with water. Discussed by Mackinnon, with colour reproduction on the cover. Colour reproduction in Pacioli, Summa, 1994 reprint supplement.

Luca Pacioli & Leonardo da Vinci. Untitled MS of 1498, beginning: Tavola dela presente opera e utilissimo compendio detto dela divina proportione dele mathematici discipline e lecto -- generally called De divina proportione. Three copies of this MS were made. One is in the Civic Library of Geneva, one is the Biblioteca Ambrosiana in Milan and the third is lost. Three modern versions of this exist.

Transcription published as Fontes Ambrosiani XXXI, Bibliothecae Ambrosianae, Milan, 1956. This was sponsored by Mediobanca as a private edition. There is a copy at University College London.

Colour facsimile of the Milan copy, Silvana Editoriale, Milano, (1982), 2nd ptg, 1986. With a separate booklet giving bibliographical details and an Introduzione di Augusto Marinoni, 20pp + covers. The booklet indicates this is Fontes Ambrosiani LXXII.

Printed version: [De] Divina proportione Opera a tutti glingegni perspicaci e curiosi necessaria Ove ciascun studioso di Philosophia: Prospectiva Pictura Sculptura: Architectura: Musica: e altre Mathematice: suavissima: sottile: e admirabile doctrina consequira: e delectarassi: cōvarie questione de secretissima scientia. Ill. by Leonardo da Vinci. Paganino de Paganini, Brescia, 1509. Facsimile in series Fontes Ambrosiana, no. XXXI, Milan, 1956; also by Editrice Dominioni, Maslianico (Como), 1967. (On f. 23r, the date of completion of the original part is printed as 1497, but both MSS have 1498.)

The printed version was assembled from three codices dating from 1497 1498 and contains the above MS with several additional items. However, the diagrams in the text are simplified and the plates are in a different order. The MS has 60 coloured plates, double sided; the printed version has 59 B&W plates, single sided. There are errors of pagination and plate numbering in both versions. On f. 3 of the printed version is a list of plates and one sees that plate LXI should be numbered LVIIII and that plates LX, LXI were omitted and were to have been a hexagonal pyramid in solid and framework views (the framework view is in the MS, but the solid pyramid is not).

NOTE. Simon Finch's Catalogue 48, item 4, describes the copy that was in the Honeyman Collection and says it has 59 printed plates of geometric figures and is unique in having two contemporary additional MS plates showing the hexagonal pyramid (numbered LX and LXI), which are given in the list of plates, but which do not appear in any other known copy. It seems that these figures were overlooked in printing and that the owner of the Honeyman copy decided to make his own versions, or, more likely, got someone to make versions in the original style. There is a framework hexagonal pyramid in the MS, and this makes it seem likely that these figures had been prepared and were omitted in printing -- indeed the Honeyman leaves could be the overlooked drawings. That leaves the question of whether there was a solid hexagonal pyramid in the MS?

Most pictures come in pairs -- a solid figure and then a framework figure. There are the five regular polyhedra, the following six Archimedean polyhedra: truncated tetrahedron, cubo-octahedron, truncated octahedron, truncated icosahedron, icosi-dodecahedron, rhombi-cubo-octahedron and also the stella octangula. There are raised or elevated versions of the tetrahedron, cube, cubo-octahedron, icosahedron, dodecahedron, icosi-dodecahedron, rhombi-cubo-octahedron. Also triangular, square, pentagonal and hexagonal prisms and tall triangular, square and pentagonal pyramids. Also a triangular pyramid not quite regular and a sphere divided into 12 sectors and 6 zones. There are also a solid sphere, a solid cylinder, a solid cone and a framework hexagonal pyramid (the last is not in the printed version). Mackinnon says they give the same seven Archimedean polyhedra as Piero, but Piero gives five or six and Pacioli & da Vinci gives six, with only four common polyhedra. Pacioli & da Vinci assert that the rhombi-cuboctahedron arises by truncating a cuboctahedron, but this is not exactly correct.

Part of the printed version is Libellus in tres partiales tractatus divisus quae corpori regularium e depēdentiū actine perscrutatiōis ..., which is an Italian translation (probably by Luca Pacioli) of Piero della Francesca's Libellus de quinque corporibus regularibus. There is debate as to whether this was actually written by Pacioli or whether Pacioli plagiarized it and whether it actually appeared in the 1509 printing or was added to a later reprinting, etc.

Davis [p. 65] says the drawings were made from models prepared by Da Vinci. Davis [p. 74] cites Summa, Part II, f. 68v, and she quotes part of it on pp. 100-101. This is also referred to by MacKinnon [p. 170] and Taylor [p. 344], neither giving details and no two of the three agreeing on what the passage means. I have not been able to make complete sense of the passage, but it seems clearly to say that in Apr 1489, Pacioli presented models of at least the regular solids to the Duke of Urbino at the palace of Pacioli's protector [Cardinal Giuliano della Rovere, later Pope Julius II] in Rome. He then says many other dependants [= variations] of the regular solids can be made, and models were made for Pietro Valletari, Bishop of Carpentras. There is no reference to the number of models, nor their material, nor to a set being given to the Cardinal, nor whether the Cardinal was present when the models were given to the Duke. Due to a missing right parenthesis, ), the sense of one statement involving 'his own hands' could mean either that Pacioli presented the models to the Duke's own hands or that Pacioli had decorated the models himself. I doubt whether there were many other solids at this time, otherwise he would have mentioned them in the Summa -- the Summa only describes four of the Archimedean solids and only two of them are in Part II. I suggest that he didn't start developing the other shapes until about 1494, or later, in 1496 when he went to Milan and met Leonardo.

However, on f. 28v of De Divina Proportione, Pacioli is clearer and says that he arranged, coloured and decorated with his own hands 60 models in Milan and two other sets for Galeazzo Sanseverino in Milan and for Piero Soderini in Florence. This refers to his time in Milan, which was 1496-1499, though the Soderini set might have been made after Pacioli and da Vinci moved to Florence.

Albrecht Dürer. Underweysung der messung .... 1525 & 1538. Op. cit. in 6.AA. Figures 29 43 (erroneously printed 34) (pp. 316-347 in The Painter's Manual, Dürer's 1525 ff. M-iii-v - N-v-r) show a net of each of the regular polyhedra, an approximate sphere (16 sectors and 8 zones), truncated tetrahedron, truncated cube, cubo-octahedron, truncated octahedron, rhombi-cubo-octahedron, snub cube, great rhombi-cubo-octahedron, polyhedron of six dodecagons and thirty-two triangles (having a pattern of four triangles replacing each triangle of the cubo-octahedron, so a sort of truncated cubo-octahedron -- not an Archimedean solid and not correctly drawn) and an elongated hexagonal bipyramid (not even regular faced). This gives 7 of the 13 non-regular Archimedean polyhedra. Mackinnon says Figures 29 41 show a net of each of the regular polyhedra and the same seven Archimedean ones as given by Pacioli & da Vinci, but they give six and there are only four common ones. In the revised version of 1538, figure 43 is replaced by the truncated icosahedron and icosi-dodecahedron (figures 43 & 43a, pp. 414 419 in The Painter's Manual), giving 9 of the 13 non-regular Archimedean solids. P. 457 shows the remaining four Archimedean cases from an 1892 edition.

Albrecht Dürer. Elementorum Geometricorum (?). 1534. Op. cit. in 6.AA. Liber quartus, fig. 29-43, pp. 145-158 shows the same material as in the 1525 edition.

See Barbaro, 1568, in 6.AT.2, pp. 45-104 for drawings and nets of 11 of the 13 Archimedean solids - he omits the two snub solids.

See Jamnitzer, 1568, in 6.AT.2 for drawings of eight of the 13 Archimedean solids.

J. Kepler. Letter to Maestlin (= Mästlin). 22 Nov 1599. In: Johannes Kepler Gesammelte Werke, ed. by M. Caspar, Beck, Munich, 1938. Vol 14, p. 87, letter 142, lines 21-22. ??NYS. Described by Field, p. 202. Describes both rhombic solids.

J. Kepler. Strena seu De Nive Sexangula [A New Year's Gift or The Six Cornered Snowflake]. Godfrey Tampach, Frankfurt am Main, 1611. (Reprinted in: Johannes Kepler Gesammelte Werke; ed. by M. Caspar, Beck, Munich, 1938, vol. 4, ??NYS.) Reprinted, with translation by C. Hardie and discussion by B. J. Mason & L. L. Whyte, OUP, 1966. I will cite the pages from Kepler (and then the OUP pages). P. 7 (10 11). Mentions 'the fourteen Archimedean solids' [sic!]. Describes the rhombic dodecahedron and mentions the rhombic triacontahedron. The translator erroneously adds that the angles of the rhombi of the dodecahedron are 6O^o and 120^o. Kepler adds that the rhombic dodecahedron fills space. Kepler's discussion is thorough and gives no references, so he seems to feel it was his own discovery.

J. Kepler. Harmonices Mundi, 1619. Book II, opp. cit. above. Prop. XXVII, p. 61. Proves that there are just two rhombic 'semi regular' solids, the rhombic dodecahedron and the rhombic triacontahedron, though the cube and the 'baby blocks' tessellation can also be considered as limiting cases. He illustrates both polyhedra. Def. XIII, p. 50 & plate (missing in facsimile). Mentions prisms and antiprisms. Prop. XXVIII, pp. 61 65. Finds the 13 Archimedean solids and illustrates them -- the first complete set -- but he does not formally show existence.

J. Kepler. Epitome Astronomiae Copernicanae. Linz, 1618-1621. Book IV, 1620. P. 464. = Johannes Kepler Gesammelte Werke; ed. by M. Caspar, Beck, Munich, 1938, vol. 7, p. 272. Shows both rhombic solids.

The following table shows which Archimedean polyhedra appear in the various early books from della Francesca to Kepler.
dF = della Francesca, c1487, folio

T indicates the object appears in the Trattato of c1480.

P = Pacioli, Summa, Part II, 1494, folio.

P&dV = Pacioli & da Vinci, 1498, plate number of the solid version; the framework version is

the next plate

D = Dürer, 1525, plate

+ indicates the object is added in the 1538 edition.

B = Barbaro, 1568, page

J = Jamnitzer, 1568, plate

K = Kepler, 1619, figure