Sources page biographical material

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

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

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

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

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

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

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

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

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

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

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

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

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

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