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THE LEAST-AREA TETRAHEDRAL TILE OF SPACE ELIOT BONGIOVANNI, ALEJANDRO DIAZ, ARJUN KAKKAR, NAT SOTHANAPHAN Abstract. We determine the least-area unit-volume tetrahedral tile of Euclidean space, without the constraint of Gallagher et al. that the tiling uses only orientation- preserving images of the tile. The winner remains Sommerville's type 4v. Contents 1. Introduction 1 2. Tetrahedra and Tilings 5 3. Previous Literature 7 4. Edge-Length Graphs 16 5. Tetrahedra with Dihedral Angles 2π=n 19 6. Classification of Tetrahedra 25 7. Previously Non-Characterized Ten Types 37 8. Isoperimetry 45 9. Code 48 10. Remaining Cases 62 11. The Best Tetrahedral Tile 69 References 71 1. Introduction arXiv:1709.04139v3 [math.MG] 6 Jun 2019 Gallagher et al. [GGH] showed that the least-surface-area, unit- volume, face-to-face, tetrahedral tile is Sommerville's [So2] type 4v, referred to in our paper as Sommerville No. 1 and illustrated in Figure 1, assuming that the tiling is orientation preserving (does not need to reflect the tile). 1 2 ELIOT BONGIOVANNI, ALEJANDRO DIAZ, ARJUN KAKKAR, NAT SOTHANAPHAN p 3 p 3 2 p 3; π=3 2; π=2 p p p 3 3 3; π=3 p p 3 3; π=3 p 3 p 2; π=2 3; π=3 p 3 2 (a) (b) Figure 1. (1a): The Sommerville No. 1 tetrahedral tile labeled with edge lengths and dihedral angles. (1b): The arrangement of three copies of the Sommerville No. 1 tile into a triangular prism, proving that it tiles space. We prove that the Sommerville No. 1 even among non-orientation- preserving tilings is still surface area minimizing. Theorem 1.1 (Thm. 11.1). The least-area, face-to-face tetrahedral tile of unit volume is uniquely the Sommerville No. 1. The least-surface-area property in question could be interpreted as being the closest one to the regular tetrahedron among all face-to-face space-tiling tetrahedra. Currently, there is no list of all tetrahedral tiles. Moreover, there are infinitely many tetrahedra that tile in a non-orientation-preserving manner, all known examples provided by Goldberg [Gol]. We charac- terize a subclass of tetrahedra that includes all possible candidates for surface area minimization and find that there are none that beat the Sommerville No. 1. First, we classify all tetrahedra into 25 types (Sect. THE LEAST-AREA TETRAHEDRAL TILE OF SPACE 3 6), fifteen with the tiles completely identified. The Sommerville No. 1 is the least-area tile among these fifteen types. The remaining ten types can be reduced to a total of 2074 cases by certain linear equations and bounds on the dihedral angles (Sects. 7{9). In particular, every dihedral angle must be greater than 36.5 degrees (Cor. 8.3). Necessary conditions for dihedral angles of tetrahedra of the given type, imple- mented by careful and rigorous computer code, reduce the cases from 2074 to seven (Sect. 9). These seven cases are eliminated by special arguments (Sect. 10), leaving no candidate to tile with less surface area than the Sommerville No. 1. Trivial Dehn invariant is a necessary but not sufficient condition for a polyhedron to tile and is therefore a potentially useful tool in narrowing down to potential tiles. Unfortunately, we were not able to find a useful way to compute Dehn invariants. As a consequence of searching for the surface-area-minimizing tetra- hedral tile without a complete list of tetrahedral tiles, this paper also completely characterizes the tiles of fifteen of 25 types and provides some necessary conditions for the remaining ten types to tile. This paper is only concerned with surface area minimization, but those in- terested in tetrahedral tiles in general may find this information useful. The known properties of all 25 types are given in Sections 6{7. Many of the methods used here, particularly the edge-length graphs described in Section 4, could be extended to tiles of other n-hedra or non-face- to-face tiles (Section 11). Even the characterization of convex planar tiles was settled only recently with Rao's [Rao] characterization of pentagonal tilings. As for general n-hedral tiles, Gallagher et al. [GGH] give conjectures for all values of n. The only previous proofs were for the cube (n = 6) and the triangular prism (n = 5). In both cases, the result for the least-area tile is simply the least-area n-hedron, which happens to tile. This paper presents the first proven case where requiring an n-hedron to tile changes the surface area minimizer. 4 ELIOT BONGIOVANNI, ALEJANDRO DIAZ, ARJUN KAKKAR, NAT SOTHANAPHAN Outline of paper. Section 2 sets terminology and summarizes previously- known facts about tetrahedra that are used in proofs throughout this paper. Section 3 gives a comprehensive review of known tetrahedral tiles and proves that the Sommerville No. 1 is the least-area tile among known tetrahedral tiles. Section 4 introduces a combinatorial object called an edge-length graph, which is used to deduce linear systems of dihedral angles that are necessary conditions for tetrahedral tiles. Section 5 completely identifies all tiling tetrahedra whose dihedral angles are all of the form 2π=n. Section 6 classifies all tetrahedra into 25 types. Of these 25 types, the tiling behavior of eleven types has been completely characterized by previous literature. An additional four types are proven not to tile, which was not previously addressed by the literature. Some necessary conditions for the remaining ten types to tile are given in Section 7. Section 8 provides a lower bound on dihedral angles that allows us to search the remaining ten types for a tetrahedral tile with less surface area than the Sommerville No. 1. Section 9 describes how computer code is used with necessary con- ditions from Sections 7 and 8 to reduce the number of candidates to beat the Sommerville No. 1 to seven remaining cases. Section 10 proves that none of these remaining seven cases tiles with less surface area than the Sommerville No. 1. Finally, in Section 11 we conclude that the Sommerville No. 1 is the least-surface-area tetrahedral tile. We comment on how methods used in this paper may extend to tilings of other polyhedra or non-face-to- face-tiles. We conjecture that the list of tetrahedral tiles identified in previous literature is exhaustive. We also conjecture that a surface- area-minimizing n-hedron is identical to its mirror image. Note on code. Many proofs in this paper are computer-assisted. The code for these computations is available in the GitHub repository at https://github.com/arjunkakkar8/Tetrahedra. THE LEAST-AREA TETRAHEDRAL TILE OF SPACE 5 Acknowledgements. The results presented in this paper came from research conducted by the 2017 Geometry Group at the Williams Col- lege SMALL NSF REU. We would like to thank our adviser Frank Mor- gan for his guidance on this project, as well as Stan Wagon [BLWW, Chapt. 4] for sending us his code on reliable computation methods used in this paper. We would also like to thank the National Science Foundation; Williams College; Michigan State University; University of Maryland, College Park; and Massachusetts Institute of Technology. 2. Tetrahedra and Tilings Section 2 sets terminology and summarizes previously known facts about tetrahedra that are used in proofs throughout this paper. We use notation consistent with previous literature. Let T = V1V2V3V4 3 be a tetrahedron where V1;V2;V3;V4 are vertices in R . For fi; j; k; lg = f1; 2; 3; 4g, let Fi be the face opposite the vertex Vi with area de- noted jFij. Let eij be the edge joining Vi and Vj with length denoted dij = jeijj. (Note that the edge between the faces Fi and Fj is ekl, not eij.) Let θij be the dihedral angle between two faces adjacent to the edge eij, i.e. between faces Fk and Fl. We assume that all tilings are face-to-face. We make a distinction between orientation-preserving and non-orientation-preserving tilings. Definition 2.1. A tiling is orientation-preserving if any two tiles are 3 equivalent under an orientation-preserving isometry of R . If a tetrahedron tiles without the use of its mirror image or if the tetrahedron is not distinct from its mirror image, then the resulting tiling is orientation-preserving. On the other hand, if a tetrahedron tiles with the use of its distinct mirror image, then the resulting tiling is non-orientation-preserving. Lemmas 2.2{2.6 summarize previously known and useful facts. Lemma 2.2 (Law of cosines for a tetrahedron). 2 2 2 2 jF1j = jF2j +jF3j +jF4j −2 (jF2j jF3j cos θ14 + jF2j jF4j cos θ13 + jF3j jF4j cos θ12) : 6 ELIOT BONGIOVANNI, ALEJANDRO DIAZ, ARJUN KAKKAR, NAT SOTHANAPHAN Lemma 2.3 relates the areas of the faces to the dihedral angles be- tween them. Geometrically, it says that the area of the base face F1 is equal to the sum of the areas of the projections of the remaining faces. Lemma 2.3. jF1j = jF2j cos θ34 + jF3j cos θ24 + jF4j cos θ23: Moreover, if cij = cos θij, then: −1 c34 c24 c23 c34 −1 c14 c13 = 0; c24 c14 −1 c12 c23 c13 c12 −1 T and the area vector (jF1j ; jF2j ; jF3j ; jF4j) is in the null space of the matrix corresponding to this determinant. The following lemmas from Wirth and Dreiding [WD2] impose con- straints on the dihedral angles and edge lengths of a tetrahedron. Lemma 2.4 ([WD2, Lemmas 1 and 3]). Ω1 = θ12 + θ13 + θ14 − π > 0; where Ω1 is the solid angle at the vertex V1 and θ13 + θ14 + θ24 + θ23 < 2π: Lemma 2.5 ([WD2, Lemma 4 and Rmk.
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