Sources page biographical material

For reference, we number the seven bars in the reverse-S pattern

shown. We can then refer to a pattern by its binary 7-tuple or its decimal equivalent. E.g. the number one is displayed by having bars 3 and 7 on, which gives a binary pattern 1000100 corresponding to decimal 68. NOTE that there is some ambiguity with the 6 / 9. Most versions use the upper / lower bar for these, i.e. 1101111 / 1111011, but the bar is sometimes omitted, giving 1001111 / 1111001. I will assume the first case unless specified.

I have been interested in these for some time for several reasons. First, my wife has such a clock on her side of the bed and she often has a glass of water in front of it, causing patterns to be reversed. At other times the clock has been on the floor upside down, causing a different reversal of patterns. Second, segments often fail or get stuck on and I have tried to analyse which would be the worst segment to fail or get stuck. As an example, the clock in my previous car went from 16:59 to 15:00. Third, I have analysed which segment(s) in a clock are used most/least often.

Birtwistle. Calculator Puzzle Book. 1978. Prob. 35: New numbers, pp. 26-27 & 83. Asks for the number of new digits one can make, subject to their being connected and full height. Says it is difficult to determine when these are distinct -- e.g. calculators differ as to the form of their 6s and 9s -- so he is not sure how to count, but he gives 22 examples. I find there are 55 connected, full-height patterns.

Gordon Alabaster, proposer & Robert Hill, solver. Problem 134.3 -- Clock watching. M500 134 (Aug 1993) 17 & 135 (Oct 1993) 14-15. Proposer notes that one segment of the units digit of the seconds on his station clock was stuck on, but that the sequence of symbols produced were all proper digits. Which segment was stuck? Asks if there are answers for 2, ..., 6 segments stuck on. Solver gives systematic tables and discusses problems of how to determine which segment(s) are stuck and whether one can deduce the correct time when the stuck segments are known.

Martin Watson. Email to NOBNET, 17 Apr 2000 08:17:32 PDT [NOBNET 2334]. Observes that the 10 digits have a total of 49 segments and asks if they can be placed on a 4 x 5 square grid. He calls these forms 'digigrams'. He had been unable to find a solution but Leonard Campbell has found 5 distinct solutions, though they do no differ greatly. He has the pieces and some discussion on his website: http://martnal.tripod.com/puzzles.html . Dario Uri [22 Apr 2000 14:44:35 +0200] found two extra solutions, but Rick Eason [22 Apr 2000 09:37: -0400] also found these, but points out that these have an error due to misreading the lattice which gives the two bars of the 1 being parallel instead of end to end. Eason's program also found the 5 solutions.
5.AD. STACKING A DECK TO PRODUCE A SPECIAL EFFECT
New section. This refers to the process of arranging a deck of cards or a stack of coins so that dealing it by some rule produces a special effect. In many cases, this is just inverting the permutation given by the rule and the Josephus problem (7.B) is a special case. Other cases involve spelling out the names of cards, etc.
Will Blyth. Money Magic. C. Arthur Pearson, London, 1926. Alternate heads, pp. 61-63. Stack of eight coins. Place one on the table and the next on the bottom of the stack. The sequence of placed coins is to alternate heads and tails. How do you arrange the stack? Answer is HHTHHTTT. This is the same process as counting out by 2s -- see 7.B.

Doubleday - 2. 1971. Heads and tails, pp. 105-106. Same as Blyth, but with six coins and solution HTTTHH.

5.AE. REVERSING CUPS
New section. There are several versions of this and they usually involve parity. The basic move is to reverse two of the cups. The classic problem seems to be to start with UDU and produce DDD in three moves. A trick version is to demonstrate this several times to someone and then leave him to start from DUD. Another easy problem is to leave three cups as they were after three moves. This is equivalent to a 3 x 3 array with an even number in each row and column -- see 6.AO.2. These problems must be much older than I have, but the following are the only examples I have yet noted.
Anonymous. Social Entertainer and Tricks (thus on spine, but running title inside is New Book of Tricks). Apparently a compilation with advertisements for Johnson Smith (Detroit, Michigan) products, c1890?. P. 38a: Bottoms up. Given UDU, produce DDD in three moves.

Young World. c1960. P. 39: Water switch. Full and empty glasses: FFFEEE. Make them alternately full and empty in one move.

Putnam. Puzzle Fun. 1978.

No. 3: Tea for three, pp. 1 & 25. Cups given as UDU. Produce DDD in three moves.

No. 16: Glass alignment, pp. 5 & 28. Six cups arranged UUUDDD. Produce an alternating row. He gets UDUDUD in three moves. I can get DUDUDU in four moves.
5.AF. SPOTTING DICE
New section. In the early 1980s, I asked Richard Guy what was the 'standard' configuration for a die and later asked Ray Bathke if he used a standard pattern. Assuming opposite sides add to seven there are two handednesses. But also the spot pattern of the two, three and six has two orientations, giving 16 different patterns of die. Ray said that when he furnished dice with games, some customers had sent them back because they weren't the same. Within about three years, I had obtained examples of all sixteen patterns! Indeed, I often found several patterns in a single batch from one manufacturer. Ray Bathke also pointed out that the small dice that come from the oriental games have the two arranged either horizontally or vertically rather than diagonally, giving another 16 patterns. I have only obtained five of these, but with both handednesses included. I used this idea in one of my Brain Twisters, cf below.

Since the 2, 3 and 6 faces all meet at a corner, one has just to describe this corner. The 2, 3, 6 can be clockwise around the corner or anti-clockwise. Note that 236 is clockwise if and only if 132 is clockwise. The position of the 2 and 3 can be described by saying whether the pattern points toward or away from the corner. If we place the 2 upward, then 6 will be a vertical face and we can describe it by saying whether the lines of three spots are vertical or horizontal. Guy told me a system for describing a die, but it's not in Winning Ways and I've forgotten it, so I'll invent my own.

We write the sequence 236 if 236 is arranged clockwise at the 236 corner and we write 263 otherwise. When looked at cornerwise, with the 2 on top, the pattern of the 2 may appear vertical or horizontal. We write 2 when it is vertical and 2 when it is horizontal. (For oriental dice, the 2 will appear on a diagonal and can be indicated by 2 or 2. If we now rotate the cube to bring the 3 on top, its pattern will appear either vertical or horizontal and we write 3 or 3. Putting the 2 back on top, the 6 face will be upright and the lines of three spots will be either vertical or horizontal, which we denote by 6 or 6.
David Singmaster. Dicing around. Weekend Telegraph (16 Dec 1989). = Games & Puzzles No. 15 (Jun 1995) 22-23 & 16 (Jul 1995) 43-44. How many dice are there? Describes the normal 16 and mentions the other 16.

Ian Stewart. The lore and lure of dice. SA (Nov 1997) ??. He asserts that the standard pattern has 132 going clockwise at a corner, except that the Japanese use the mirror-image version in playing mah-jongg. His picture has both 2 and 3 toward the 236 corner and the 6 being vertical, i.e. in pattern 236. He discusses crooked dice of various sorts and that the only way to make all values from 1 to 12 equally likely is to have 123456 on one die and 000666 on the other.

Ricky Jay. The story of dice. The New Yorker (11 Dec 2000) 90-95.
5.AG. RUBIK'S CUBE AND SIMILAR PUZZLES
I have previously avoided this as being too recent to be covered in a historical work, but it is now old enough that it needs to be covered, and there are some older references. Much of the history is given in my Notes on Rubik's Cube and my Cubic Circular. Jaap Scherphuis has sent me a file of puzzle patents and several dozen of them could be entered here, but I will only enter older or novel items. Scherphuis's file has about a dozen patents for the 4 x 4 x 4 and 5 x 5 x 5 cubes! See Section 5.A for predecessors of the idea. However, this Section will mostly deal with puzzles where pieces are permuted without having any empty places, so these are generally permutation puzzle.
5.AG.1. RUBIK'S CUBE

New section. Much to be added.

Richard E. Korf. Finding optimal solutions to Rubik's Cube using pattern databases. Proc. Nat. Conf. on Artificial Intelligence (AAAI-97), Providence, Rhode Island, Jul 1997, pp. 700-705. Studies heuristic methods of finding optimal solutions of the Cube. Claims to be the first to find optimal solutions for random positions of the Cube -- but I think others such as Kociemba and Reid were doing it up to a decade earlier. For ten random examples, he found optimal solutions took 16 moves in one case, 17 moves in three cases, 18 moves in six cases, from which he asserts the median optimal solution length seems to be 18. He uses the idea of axial moves and obtains the lower bound of 18 for God's Algorithm, as done in my Notes in 1980. Cites various earlier work in the field, but only one reference to the Cube literature.

Richard E. Korf & Ariel Felner. Disjoint pattern database heuristics. Artificial Intelligence 134 (2002) 9-22. Discusses heuristic methods of solving the Fifteen Puzzle, Rubik's Cube, etc. Asserts the median optimal solution length for the Cube is only 18. Seems to say one of the problems in the earlier paper took a couple of weeks running time, but improved methods of Kociemba and Reid can find optimal solutions in about an hour.

5.AG.2. HUNGARIAN RINGS, ETC.
New section. Much to be added.
William Churchill. US Patent 507,215 -- Puzzle. Applied: 28 May 1891; patented: 24 Oct 1893. 1p + 1p diagrams. Two rings of 22 balls, intersecting six spaces apart.

Hiester Azarus Bowers. US Patent 636,109 -- Puzzle. Filed: 16 Aug 1899; patented 31 Oct 1899. 2pp + 1p diagrams. 4 rotating discs which overlap in simple lenses.

Ivan Moscovich. US Patent 4,509,756 -- Puzzle with Elements Transferable Between Closed loop Paths. Filed: 18 Dec 1981; patented: 9 Apr 1985. Cover page + 3pp + 2pp diagrams. Two rings of 18 balls, each stretched to have two straight sections with semicircular ends. The rings cross in four places, at the ends of the straight sections, so adjacent crossing points are separated by two balls. I'm not sure this was ever produced. Mentions three circular rings version, but there each pair of rings only overlaps in two places so this is a direct generalization of the Hungarian Rings.

David Singmaster. Hungarian Rings groups. Bull. Inst. Math. Appl. 20:9/10 (Sep/Oct 1984) 137-139. [The results were stated in Cubic Circular 5 & 6 (Autumn & Winter 1982) 9 10.] An article by Philippe Paclet [Des anneaux et des groupes; Jeux et Stratégie 16 (Aug/Sep 1982) 30-32] claimed that all puzzles of two rings have groups either the symmetric or the alternating group on the number of balls. This article shows this is false and determines the group in all cases. If we have rings of size m, n and the intersections are distances a, b apart on the two rings. Then the group, G(m, n, a, b) is the symmetric group on m+n-2 if mn is even and is the alternating group if mn is odd; except that G(4, 4, 1, 1) is the exceptional group described in R. M. Wilson's 1974 paper: Graph puzzles, homotopy and the alternating group -- cited in Section 5.A under The Fifteen Puzzle -- and is also the group generated by two adjacent faces on the Rubik Cube acting on the six corners on those faces; and except that G(2a, 2b, a, b) keeps antipodal pairs at antipodes and hence is a subgroup of the wreath product Z₂ wr S_a+b 1, with three cases depending on the parities of a and b.

Bala Ravikumar. The Missing Link and the Top-Spin. Report TR94-228, Department of Computer Science and Statistics, University of Rhode Island, Jan 1994. Top-Spin has a cycle of 20 pieces and a small turntable which permits inverting a section of four pieces. After developing the group theory and doing the Fifteen Puzzle and the Missing Link, he shows the state space of Top-Spin is S₂₀.
6. GEOMETRIC RECREATIONS
6.A. PI
This is too big a topic to cover completely. The first items should be consulted for older material and the general history. Then I include material of particular interest. See also 6.BL which has some formulae which are used to compute π. I have compiled a separate file on the history of π.
Augustus De Morgan. A Budget of Paradoxes. (1872); 2nd ed., edited by D. E. Smith, (1915), Books for Libraries Press, Freeport, NY, 1967.

J. W. Wrench Jr. The evolution of extended decimal approximations to π. MTr 53 (Dec 1960) 644 650. Good survey with 55 references, including original sources.

Petr Beckmann. A History of π. The Golem Press, Boulder, Colorado, (1970); 2nd ed., 1971.

Lam Lay-Yong & Ang Tian-Se. Circle measurements in ancient China. HM 13 (1986) 325 340. Good survey of the calculation of π in China.

Dario Castellanos. The ubiquitous π. MM 61 (1988) 67-98 & 148-163. Good survey of methods of computing π.

Joel Chan. As easy as pi. Math Horizons 1 (Winter 1993) 18-19. Outlines some recent work on calculating π and gives several of the formulae used.

David Singmaster. A history of π. M500 168 (Jun 1999) 1-16. A chronology. (Thanks to Tony Forbes and Eddie Kent for carefully proofreading and amending my file.)
Aristophanes. The Birds.  414. Lines 1001 1005. In: SIHGM I 308 309. Refers to 'circle-squarers', possibly referring to the geometer/astronomer Meton.
E. J. Goodwin. Quadrature of the circle. AMM 1 (1894) 246 247.

House Bill No. 246, Indiana Legislature, 1897. "A bill for an act introducing a new mathematical truth ..." In Edington's paper (below), p. 207, and in several of the newspaper reports.

(Indianapolis) Journal (19 Jan 1897) 3. Mentions the Bill in the list of bills introduced.

Die Quadratur des Zirkels. Täglicher Telegraph (Indianapolis) (20 Jan 1897) ??. Surveys attempts since -2000 and notes that Lindemann and Weierstrass have shown that the problem is impossible, like perpetual motion.

A man of 'genius'. (Indianapolis) Sun (6 Feb 1897) ??. An interview with Goodwin, who says: "The astronomers have all been wrong. There's about 40,000,000 square miles on the surface of this earth that isn't here." He says his results are revelations and gives several rules for the circle and the sphere.

Mathematical Bill passed. (Indianapolis) Journal (6 Feb 1897) 5. "This is the strangest bill that has ever passed an Indiana Assembly." Gives whole text of the Bill.

Dr. Goodwin's theaorem (sic) Resolution adopted by the House of Representatives. (Indianapolis) News (6 Feb 1897) 4. Gives whole text of the Bill.

The Mathematical Bill Fun-making in the Senate yesterday afternoon -- other action. (Indianapolis) News (13 Feb 1897) 11. "The Senators made bad puns about it, ...." The Bill was indefinitely postponed.

House Bills in the Senate. (Indianapolis) Sentinel (13 Feb 1897) 2. Reports the Bill was killed.

(No heading??) (Indianapolis) Journal (13 Feb 1897) 3, col. 4. "... indefinitely postponed, as not being a subject fit for legislation."

Squaring the circle. (Indianapolis) Sunday Journal (21 Feb 1897) 9. Says Goodwin has solved all three classical impossible problems. Says π = 3.2, using the fact that 2 = 10/7, giving diagrams and a number of rules.

My thanks to Underwood Dudley for locating and copying the above newspaper items.

C. A. Waldo. What might have been. Proc. Indiana Acad. Science 26 (1916) 445 446.

W. E. Edington. House Bill No. 246, Indiana State Legislature, 1897. Ibid. 45 (1935) 206 210.

A. T. Hallerberg. House Bill No. 246 revisited. Ibid. 84 (1975) 374 399.

Manuel H. Greenblatt. The 'legal' value of pi, and some related mathematical anomalies. American Scientist 53 (Dec 1965) 427A 434A. On p. 427A he tries to interpret the bill and obtains three different values for π.

David Singmaster. The legal values of pi. Math. Intell. 7:2 (1985) 69 72. Analyses Goodwin's article, Bill and other assertions to find 23 interpretable statements giving 9 different values of π !

Underwood Dudley. Mathematical Cranks. MAA, 1992. Legislating pi, pp. 192-197.

C. T. Heisel. The Circle Squared Beyond Refutation. Published by the author, 657 Bolivar Rd., Cleveland, Ohio, 1st ed., 1931, printed by S. J. Monck, Cleveland; 2nd ed., 1934, printed by Lezius Hiles Co., Cleveland, ??NX + Supplement: "Fundamental Truth", 1936, ??NX, distributed by the author from 2142 Euclid Ave., Cleveland. This is probably the most ambitious publication of a circle-squarer -- Heisel distributed copies all around the world.
Underwood Dudley. π_t: 1832-1879. MM 35 (1962) 153-154. He plots 45 values of π as a function of time over the period 1832-1879 and finds the least-squares straight line which fits the data, finding that π_t = 3.14281 + .0000056060 t, for t measured in years AD. Deduces that the Biblical value of 3 was a good approximation for the time and that Creation must have occurred when π_t = 0, which was in -560,615.

Underwood Dudley. π_t. JRM 9 (1976-77) 178 & 180. Extends his previous work to 50 values of π over 1826-1885, obtaining π_t = 4.59183 - .000773 t. The fact that π_t is decreasing is worrying -- when π_t = 1, all circles will collapse into straight lines and this will certainly be the end of the world, which is expected in 4646 on 9 Aug at 20:55:33 -- though this is only the expected time and there is considerable variation in this prediction. [Actually, I get that this should be on 11 Aug. However, it seems to me that circles will collapse once π_t = 2, as then the circumference corresponds to going back and forth along the diameter. This will occur when t = 3352.949547, i.e. in 3352, on 13 Dec at 14:01:54 -- much earlier than Dudley's prediction, so start getting ready now!]

6.B. STRAIGHT LINE LINKAGES
See Yates for a good survey of the field.
James Watt. UK Patent 1432 -- Certain New Improvements upon Fire and Steam Engines, and upon Machines worked or moved by the same. Granted: 28 Apr 1784; complete specification: 24 Aug 1784. 14pp + 1 plate. Pp. 4-6 & Figures 7 12 describe Watt's parallel motion. Yates, below, p. 170 quotes one of Watt's letters: "... though I am not over anxious after fame, yet I am more proud of the parallel motion than of any other invention I have ever made."

P. F. Sarrus. Note sur la transformation des mouvements rectilignes alternatifs, en mouvements circulaires; et reciproquement. C. R. Acad. Sci. Paris 36 (1853) 1036 1038. 6 plate linkage. The name should be Sarrus, but it is printed Sarrut on this and the following paper.

Poncelet. Rapport sur une transformation nouvelle des mouvements rectilignes alternatifs en mouvements circulaires et reciproquement, par Sarrut. Ibid., 36 (1853) 1125 1127.

A. Peaucellier. Lettre au rédacteur. Nouvelles Annales de Math. (2) 3 (1864) 414 415. Poses the problem.

A. Mannheim. Proces Verbaux des sceances des 20 et 27 Juillet 1867. Bull. Soc. Philomathique de Paris (1867) 124 126. ??NYS. Reports Peaucellier's invention.

Lippman Lipkin. Fortschritte der Physik (1871) 40 ?? ??NYS

L. Lipkin. Über eine genaue Gelenk Geradführung. Bull. Acad. St. Pétersbourg [=? Akad. Nauk, St. Petersburg, Bull.] 16 (1871) 57 60. ??NYS

L. Lipkin. Dispositif articulé pour la transformation rigoureuse du mouvement circulaire en mouvement rectiligne. Revue Univers. des Mines et de la Métallurgie de Liége 30:4 (1871) 149 150. ??NYS. (Now spelled Liège.)

A. Peaucellier. Note sur un balancier articulé a mouvement rectiligne. Journal de Physique 2 (1873) 388 390. (Partial English translation in Smith, Source Book, vol. 2, pp. 324 325.) Says he communicated it to Soc. Philomath. in 1867 and that Lipkin has since also found it. There is also an article in Nouv. Annales de Math. (2) 12 (1873) 71 78 (or 73?), ??NYS.

E. Lemoine. Note sur le losange articulé du Commandant du Génie Peaucellier, destiné a remplacer le parallélogramme de Watt. J. de Physique 2 (1873) 130 134. Confirms that Mannheim presented Peaucellier's cell to Soc. Philomath. on 20 Jul 1867. Develops the inversive geometry of the cell.

[J. J. Sylvester.] Report of the Annual General Meeting of the London Math. Soc. on 13 Nov 1873. Proc. London Math. Soc. 5 (1873) 4 & 141. On p. 4 is: "Mr. Sylvester then gave a description of a new instrument for converting circular into general rectilinear motion, and into motion in conics and higher plane curves, and was warmly applauded at the close of his address." On p. 141 is an appendix saying that Sylvester spoke "On recent discoveries in mechanical conversion of motion" to a Friday Evening's Discourse at the Royal Institution on 23 Jan 1874. It refers to a paper 20 pages long but is not clear if or where it was published.

H. Hart. On certain conversions of motion. Cambridge Messenger of Mathematics 4 (1874) 82 88 and 116 120 & Plate I. Hart's 5 bar linkage. Obtains some higher curves.

A. B. Kempe. On some new linkages. Messenger of Mathematics 4 (1875) 121 124 & Plate I. Kempe's linkages for reciprocating linear motion.

H. Hart. On two models of parallel motions. Proc. Camb. Phil. Soc. 3 (1876 1880) 315 318. Hart's parallelogram (a 5 bar linkage) and a 6 bar one.

V. Liguine. Liste des travaux sur les systèms articulés. Bull. d. Sci. Math. 18 (or (2) 7) (1883) 145 160. ??NYS   cited by Kanayama. Archibald; Outline of the History of Mathematics, p. 99, says Linguine is entirely included in Kanayama.

Gardner D. Hiscox. Mechanical Appliances Mechanical Movements and Novelties of Construction. A second volume to accompany his previous Mechanical Movements, Powers and Devices. Norman W. Henley Publishing Co, NY, (1904), 2nd ed., 1910. This is filled with many types of mechanisms. Pp. 245-247 show five straight-line linkages and some related mechanisms.

(R. Kanayama). (Bibliography on linkages. Text in Japanese, but references in roman type.) Tôhoku Math. J. 37 (1933) 294 319.

R. C. Archibald. Bibliography of the theory of linkages. SM 2 (1933 34) 293 294. Supplement to Kanayama.

Robert C. Yates. Geometrical Tools. (As: Tools; Baton Rouge, 1941); revised ed., Educational Publishers, St. Louis, 1949. Pp. 82-101 & 168-191. Gets up to outlining Kempe's proof that any algebraic curve can be drawn by a linkage.

R. H. Macmillan. The freedom of linkages. MG 34 (No. 307) (Feb 1960) 26 37. Good survey of the general theory of linkages.

Michael Goldberg. Classroom Note 312: A six plate linkage in three dimensions. MG 58 (No. 406) (Dec 1974) 287 289.
6.C. CURVES OF CONSTANT WIDTH
Such curves play an essential role in some ways to drill a square hole, etc.
L. Euler. Introductio in Analysin Infinitorum. Bousquet, Lausanne, 1748. Vol. 2, chap. XV, esp. § 355, p. 190 & Tab. XVII, fig. 71. = Introduction to the Analysis of the Infinite; trans. by John D. Blanton; Springer, NY, 1988-1990; Book II, chap. XV: Concerning curves with one or several diameters, pp. 212-225, esp. § 355, p. 221 & fig. 71, p. 481. This doesn't refer to constant width, but fig. 71 looks very like a Reuleaux triangle.

L. Euler. De curvis triangularibus. (Acta Acad. Petropol. 2 (1778(1781)) 3 30) = Opera Omnia (1) 28 (1955) 298 321. Discusses triangular versions.

M. E. Barbier. Note sur le problème de l'aiguille et jeu du joint couvert. J. Math. pures appl. (2) 5 (1860) 273 286. Mentions that perimeter = π * width.

F. Reuleaux. Theoretische Kinematik; Vieweg, Braunschweig, 1875. Translated: The Kinematics of Machinery. Macmillan, 1876; Dover, 1964. Pp. 129 147.

Gardner D. Hiscox. Mechanical Appliances Mechanical Movements and Novelties of Construction. A second volume to accompany his previous Mechanical Movements, Powers and Devices. Norman W. Henley Publishing Co, NY, (1904), 2nd ed., 1910.

Item 642: Turning a square by circular motion, p. 247. Plain face, with four pins forming a centred square, is turned by the lathe. A triangular follower is against the face, so it is moved in and out as a pin moves against it. This motion is conveyed by levers to the tool which moves in and out against the work which is driven by the same lathe.

Item 681: Geometrical boring and routing chuck, pp. 257-258. Shows it can make rectangles, triangles, stars, etc. No explanation of how it works.

Item 903A: Auger for boring square holes, pp. 353-354. Uses two parallel rotating cutting wheels.

H. J. Watts. US Patents 1,241,175 7 -- Floating tool chuck; Drill or boring member; Floating tool chuck. Applied: 30 Nov 1915; 1 Nov 1916; 22 Nov 1916; all patented: 25 Sep 1917. 2 + 1, 2 + 1, 4 + 1 pp + pp diagrams. Devices for drilling square holes based on the Reuleaux triangle.

T. Bonnesen & W. Fenchel. Theorie der konvexen Körper. Berlin, 1934; reprinted by Chelsea, 1971. Chap. 15: Körper konstanter Breite, pp. 127 141. Surveys such curves with references to the source material.

G. D. Chakerian & H. Groemer. Convex bodies of constant width. In: Convexity and Its Applications; ed. by Peter M. Gruber & Jörg M. Wills; Birkhäuser, Boston, 1983. Pp. 49 96. (??NYS -- cited in MM 60:3 (1987) 139.) Bibliography of some 250 items since 1930.