III IIHIIIHIII USOO5524396A United States Patent (19 11) Patent Number: 5,524,396 Lalvani 45 Date of Patent: Jun. 11, 1996 54). SPACE STRUCTURES WITH NON-PERIODIC Regular Polytopes by H. S. M. Coxeter, Dover, 1963, pp. SUBDIVISIONS OF POLYGONAL FACES 27-33 T6) Inventor: Haresh Lalvani, 164 Bank St., Apt. Primary Examiner Wynn E. Wood 2B, New York, N.Y. 10014 (57) ABSTRACT (21) Appi. No. 75,145 A family of Space structures having subdivided faces, where such faces are subdivided into rhombii in non-periodic 22 Filed: Jun. 10, 1993 arrangements. The rhombii are derived from regular planar (51) Int. Cl. ................................................ E04B 1/32 Stars with n vectors, and the source space structures are 52 U.S. Cl. ........................... 52/81.1; 52/81.2: 52/311.2: composed of regular polygons. The family includes: glo 52/DIG. 10 bally symmetric structures where the fundamental region is 58 Field of Search .................... 52/81.2, 311, DIG. 10, subdivided non-periodically, or globally asymmetric struc 52/81.1 tures composed of regular polygons which are subdivided non-periodically or asymmetrically. The rhombii can be 56 References Cited further subdivided periodically or non-periodically. The family further includes all regular polyhedra in the plane U.S. PATENT DOCUMENTS faced and curve-faced States, regular tessellations, various 1,976, 188 9/1934 Hozawa ................................... 52(81.2 curved polygons, cylinders and toroids, curved space laby 2,361,540 10/1944. Forbes ......... 52/82 rinths, and regular structures in higher-dimensional and 2,978,704 4/1961 Cohen et al. 52/812 hyperbolic space. The structures can be isolated structures or 3,058,550 971962 Richter ..................................... 52A812 grouped to fill space. Applications include architectural 3,154,887 11/1964 Schmidt ................................... 52/812 space structures, fixed or retractable space frames, domes, 3,722,153 3/1973 Baer. vaults, saddle structures, plane or curved tiles, model-kits, 4,133,152 1/1979 Penrose toys, games, and artistic and sculptural works realized in 2 5,007,220 4/1991 Lalvani. and 3-dimensions. The structures could be composed of 5,036,635 8/1991 Lalvani ......................................... 52/8 individual units capable of being assembled or disas OTHER PUBLICATIONS sembled, or structures which are cast in one piece, or combination of both. Various tensile and compressive struc Uniform Ant-hills in the World of Golden Isozonohedra by tural systems, and techniques of triangulation could be used K. Miyazaki, Structural Topology, #4, 1980 pp. 21-30 as needed for stability. (Canada). Fourfield: Computers, Art & the 4th Dimension by Tony Robbin, Bullfinch Press, 1992 pp. 81-94. 25 Claims, 57 Drawing Sheets U.S. Patent Jun. 11, 1996 Sheet 1 of 57 5,524.396 2 3 4. 5 6 7 8 9 1C) . ?h (9d) (6O) (45) (36) (3C5) (25.7) (252) (20) (18) . (7/5 1 5 2 A / A/A 747 - - 2-2 2-3 / 2-4 1 2-5 1 2-6 ? ? 20 2f 22 A 7 ses, 3-3 3-4 3-5 / / / 22 24- 25 4. 4-5 4-6 4-4 / 26 5-5 N--3ig. 1 U.S. Patent Jun. 11, 1996 Sheet 2 of 57 5,524.396 3ig. 2. U.S. Patent Jun. 11, 1996 Sheet 3 of 57 5,524.396 n=4 nic n=6 n=7 . a.A. / is 2-34Cld -a, -223 - ?2-15:34 -:(-1: it 44 60 U.S. Patent Jun. 11, 1996 Sheet 4 of 57 5,524.396 U.S. Patent Jun. 11, 1996 Sheet 5 of 57 5,524.396 U.S. Patent Jun. 11, 1996 Sheet 6 of 57 5,524,396 U.S. Patent Jun. 11, 1996 Sheet 7 of 57 5,524,396 U.S. Patent Jun. 11, 1996 Sheet 8 of 57 5,524,396 27-A & KO/kXOkk A 2 / . 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CSS) (KS,K2 -X) C Q(SS)NQ -- 12, Sysi?63. 427 Sty \63.N3A).<>(CSS y U.S. Patent Jun. 11, 1996 Sheet 19 of 57 5,524,396 U.S. Patent Jun. 11, 1996 Sheet 20 of 57 5,524,396 S.S. 8.t $7 3. ise3.11 4KOY U.S. Patent Jun. 11, 1996 Sheet 21 of 57 5,524,396 U.S. Patent Jun. 11, 1996 Sheet 22 of 57 5,524,396 \ 2. Ns NSSS 22. AS D SSSSSN VDs ()A. k1 (S4. ROE) [N]?A U.S. Patent Jun. 11, 1996 Sheet 23 of 57 5,524,396 (KYA, KEX,E)& EXA ASXCXX SK2S2,PKKSKKissFS K*. 323s. As AS VSYNEX,AY37 Sk1NAA VXXXSS NYP3XYSyNANA N4&Y / \le 392 U.S. Patent Jun. 11, 1996 Sheet 27 of 57 5,524,396 U.S. Patent Jun. 11, 1996 Sheet 28 of 57 4% Na as KAANY6NZSS gS(XXXSSX NS2. A/13/1x3N(sax/SY YXOSVAX7 3. KYYXSS D. SSX2 &gate2XCX X 2e. U.S. Patent Jun. 11, 1996 Sheet 30 of 57 5,524.396 U.S. Patent Jun. 11, 1996 Sheet 31 of 57 5,524.396 U.S. Patent Jun. 11, 1996 Sheet 32 of 57 U.S. Patent Jun. 11, 1996 Sheet 33 of 57 5,524.396 U.S. Patent Jun. 11, 1996 Sheet 34 of 57 5,524,396 as 2-, 3-, 4-, 5-, 6-; 7: ...-- q=7 32,7: 3,7: 34,7: 35,7: 36,7: 37,7: ...-a,7: q=6 32,6: 33,6: 34,6: 35,6: 36,6: 37,6: ...3-2,68 q=2 32,2: 35,2: 34,2: 35,28 36,2: 37,28 .328 C p=2 p=3 p=4 p=5 p=6 p=7 p=ed N -- 3ig. 14 U.S. Patent Jun. 11, 1996 Sheet 35 of 57 5,524,396 2,2%22 U.S. Patent Jun. 11, 1996 Sheet 36 of 57 5,524,396 CEO.NC227 s2. /20 K2SX2SS STS-334 CSXSSX/ES)KKXXXXXXX: 13. XSSSI fixes29SC (NASA ISSx afé S.SN 120 Ax7ckAa 227& NSSNSXkNx\\ SS KKKXXXXXCCK)AYS3KNA CKCSY. If S3:2)VS X 4N X KX al-as-as U.S. Patent Jun. 11, 1996 Sheet 37 of 57 5,524,396 (LWNY 7 3. Y ' U.S. Patent Jun. 11, 1996 Sheet 39 of 57 5,524.396 U.S. Patent Jun. 11, 1996 Sheet 41 of 57 5,524,396 U.S. Patent Jun. 11, 1996 Sheet 42 of 57 5,524.396 U.S. Patent Jun. 11, 1996 Sheet 43 of 57 5,524,396 SS 23S state2a33a Seas A. s . is&SSSSSSS SESS ASSES Si:SSES. SSS& &SisSS: SESSESSSE&S SaSSSshalf SSS $3.Sks U.S. Patent Jun. 11, 1996 Sheet 44 of 57 5,524,396 43/3s SK23, 2, E3:33 &S 33S.S.S.). 3. SSS.ly NN y (a), KZS35 2-YNSDTS 3, (OKX 7 S.S.S.St. 53CE,S 33.(K)2232 a U.S. Patent Jun. 11, 1996 Sheet 45 of 57 5,524,396 ZX S/NADS. CSDASYYYNK ak SkySSS. {A AXSNS NN t N CN SkyAKN5 See Sri Ariy K : X XX:X:S A. ET N73.<KXC-KSKSSXKNSKC. KykkXXYY77 YSAYAYY-SA 2KKKNW (YA 2KSX52AX7), YERRCERYE^2 KYSCKCKYYXNE, SEE YXY. Y&2 (APYK-2 5. XY.Y., S.K..) NNke SNKNSESS.A.Y.E.E. Aye A.E.IZXXEWAD S.K.3 A. 22:52,W, SSSSSSSS&AXA2NNN 3. A33A2575A. 77 SN32.25)WN N3 (Y42A225.4% 224 NŠS SSSSSS22.52 52.2%7 2252A2a2.426 U.S. Patent Jun. 11, 1996 Sheet 48 of 57 5,524,396 U.S. Patent Jun. 11, 1996 Sheet 49 of 57 5,524,396 A. 3. (i. 2:7 A. A.C 8 2. s V N s N K KSS SSSeSN s SS2.S2 ?: 3ig. 22a. U.S. Patent Jun. 11, 1996 Sheet 51 of 57 5,524,396 SSSSS-2262s2237S42k2E2223.4%ly 12a. ŠS226233632.23SNSSS2,611252,22NSSNSSK222-23 2335 SNSSNS42(xstas?.2%A. N- 62.32232 NS-232227SS23222326 SŠ22NS22s Sea NSS3? se Syses.-, O f f2 3. K 6.KS 8E. SSE 2 (A W 2 22, (SSEE p 22 VNW 4 2 3. 3. 2x5 Á2. 22 t S 2. SyNSSS) SY (SSSSSSSSSS S NS& si. 2 222s2SSNN 22a seas U.S. Patent Jun. 11, 1996 Sheet 52 of 57 5,524,396 U.S. Patent Jun. 11, 1996 Sheet 54 of 57 5,524,396 3.g. 25l. U.S. Patent Jun. 11, 1996 Sheet 55 of 57 5,524,396 3.g. 26 U.S. Patent Jun. 11, 1996 Sheet 56 of 57 5,524,396 5,524,396 1. 2 SPACE STRUCTURES WITH NON-PERIODC The structual advantages of the 'new' space structures SUBDIVISIONS OF POLYGONAL FACES disclosed here remain to be examined and analyzed.