Three dimensional geometry, ZOME, and the elusive tetrahedron

Transcription

1 Three dimensional geometry, ZOME, and the elusive tetrahedron N J Wildberger UNSW July 2012 N J Wildberger () Tetrahedron trigonometry July / 36

2 Overview A) Introduce basic ideas of affi ne and projective rational trigonometry, with some proofs of main laws N J Wildberger () Tetrahedron trigonometry July / 36

3 Overview A) Introduce basic ideas of affi ne and projective rational trigonometry, with some proofs of main laws B) Explain the ZOME construction system, and list primitive ZOME triangles N J Wildberger () Tetrahedron trigonometry July / 36

4 Overview A) Introduce basic ideas of affi ne and projective rational trigonometry, with some proofs of main laws B) Explain the ZOME construction system, and list primitive ZOME triangles C) Discuss a particular ZOME tetrahedron arising from the dodecahedron N J Wildberger () Tetrahedron trigonometry July / 36

5 Overview A) Introduce basic ideas of affi ne and projective rational trigonometry, with some proofs of main laws B) Explain the ZOME construction system, and list primitive ZOME triangles C) Discuss a particular ZOME tetrahedron arising from the dodecahedron D) Develop some of the main laws of three dimensional trigonometry (study of a general tetrahedron) N J Wildberger () Tetrahedron trigonometry July / 36

6 Overview A) Introduce basic ideas of affi ne and projective rational trigonometry, with some proofs of main laws B) Explain the ZOME construction system, and list primitive ZOME triangles C) Discuss a particular ZOME tetrahedron arising from the dodecahedron D) Develop some of the main laws of three dimensional trigonometry (study of a general tetrahedron) E) Illustrate basic laws and give some projects/challenges. N J Wildberger () Tetrahedron trigonometry July / 36

7 Overview A) Introduce basic ideas of affi ne and projective rational trigonometry, with some proofs of main laws B) Explain the ZOME construction system, and list primitive ZOME triangles C) Discuss a particular ZOME tetrahedron arising from the dodecahedron D) Develop some of the main laws of three dimensional trigonometry (study of a general tetrahedron) E) Illustrate basic laws and give some projects/challenges. History: Aristotle, Plato, Euclid, Tartaglia, Euler, Gauss, Cayley, Menger, Monge, Wolstenholme, Hilbert, Dehn, G. Richardson (The Trigonometry of the Tetrahedron, Math. Gazette, 1902) ++ N J Wildberger () Tetrahedron trigonometry July / 36

8 The elusive tetrahedron P 1 P 2 P 3 P 4 In classical trigonometry one tries to determine the various relations between the 6 side lengths, the 12 face angles, the 6 dihedral edge angles (between faces), the 4 solid angles, and the volume. There are also other secondary quantities. It is probably fair to say this has nowhere been done completely. With rational trigonometry we start with quadrances Q ij = Q (P i, P j ) P i P j P i P j instead of distances, but the real innovation is replacing angles with spreads. N J Wildberger () Tetrahedron trigonometry July / 36

9 The basic quantities: quadrances and spreads Affi ne rational trigonometry of a face + projective rational trigonometry of a tripod N J Wildberger () Tetrahedron trigonometry July / 36

10 Quadrance between points We start with a dot/inner product on a vector/linear space, such as (x 1, y 1 ) (x 2, y 2 ) = x 1 x 2 + y 1 y 2 or in three dimensions (x 1, y 1, z 1 ) (x 2, y 2, z 2 ) = x 1 x 2 + y 1 y 2 + z 1 z 2. The quadrance of a vector v is Q ( v ) v v. The quadrance between points A 1 and A 2 is ( ) Q (A 1, A 2 ) Q A 1 A 2 so that Q ([x 1, y 1 ], [x 2, y 2 ]) = (x 2 x 1 ) 2 + (y 2 y 1 ) 2. N J Wildberger () Tetrahedron trigonometry July / 36

11 Spread between lines In rational trigonometry we avoid the transcendental aspects of angles and the circular functions and their inverse functions. The spread s between vectors v and w is s ( v, w ) 1 ( v w ) 2 Q ( v ) Q ( w ) = 1 ( v w ) 2 ( v v ) ( w w ). So s ((x 1, y 1 ), (x 2, y 2 )) = 1 (x 1x 2 + y 1 y 2 ) 2 (x y 2 1 ) (x y 2 2 ) = (x 1y 2 x 2 y 1 ) 2 (x y 2 1 ) (x y 2 2 ). This extends to a notion of spread between lines (not rays!!) N J Wildberger () Tetrahedron trigonometry July / 36

12 Laws of affi ne rational trigonometry Triangle A 1 A 2 A 3 with quadrances Q 1, Q 2, Q 3 and spreads s 1, s 2, s 3 : Cross law (Q 1 + Q 2 Q 3 ) 2 = 4Q 1 Q 2 (1 s 3 ) Spread law Triple spread formula s 1 Q 1 = s 2 Q 2 = s 3 Q 3 (s 1 + s 2 + s 3 ) 2 = 2 ( s s s 2 3 ) + 4s1 s 2 s 3 N J Wildberger () Tetrahedron trigonometry July / 36

13 Proofs of main laws: Cross law so Thus ( ) ( Q 3 = Q A 1 A 2 = A 3 A 2 A ) 3 A 1 ( A 3 A 2 A ) 3 A 1 = Q 1 + Q 2 2 s 3 = 1 ( A 3 A 2 A ) 2 3 A 1 ( A 3 A 2 A ) 3 A 1 Q 1 Q 2 = 1 (Q 1 + Q 2 Q 3 ) 2 4Q 1 Q 2 (Q 1 + Q 2 Q 3 ) 2 = 4Q 1 Q 2 (1 s 3 ). Cross law. N J Wildberger () Tetrahedron trigonometry July / 36

14 Proofs of main laws: Spread law and quadrea From the Cross law (Q 1 + Q 2 Q 3 ) 2 = 4Q 1 Q 2 (1 s 3 ) we get By symmetry so that 4Q 1 Q 2 s 3 = 4Q 1 Q 2 (Q 1 + Q 2 Q 3 ) 2 = (Q 1 + Q 2 + Q 3 ) 2 2 ( Q1 2 + Q2 2 + Q3 2 ) A = quadrea of triangle = 16 (area) 2 4Q 1 Q 2 s 3 = 4Q 1 Q 3 s 2 = 4Q 2 Q 3 s 1 = A s 1 Q 1 = s 2 Q 2 = s 3 Q 3 = A 4Q 1 Q 2 Q 3. Spread law. N J Wildberger () Tetrahedron trigonometry July / 36

15 Proofs of main laws: Triple spread formula From the Spread law s 1 Q 1 = s 2 Q 2 = s 3 Q 3 1 D so that Q 1 = Ds 1, Q 2 = Ds 2 and Q 3 = Ds 3. Sub into the Cross law (Q 1 + Q 2 Q 3 ) 2 = 4Q 1 Q 2 (1 s 3 ) to get or Thus D 2 (s 1 + s 2 s 3 ) 2 = 4D 2 s 1 s 2 (1 s 3 ) (s 1 + s 2 s 3 ) 2 = 4s 1 s 2 (1 s 3 ). (s 1 + s 2 + s 3 ) 2 = 2 ( s s s 2 3 ) + 4s1 s 2 s 3. TSF. N J Wildberger () Tetrahedron trigonometry July / 36

16 Projective rational trigonometry A projective point a = [x : y : z] is a one dimensional subspace of V 3, a three dim vector space over a field (usually the rationals). A projective line L = (l : m : n) is a two dimensional subspace of V 3, representing the plane lx + my + nz = 0. The projective quadrance q (a 1, a 2 ) is defined to be the spread between the projective points a 1, a 2. If a 1 = [x 1 : y 1 : z 1 ] and a 2 = [x 2 : y 2 : z 2 ] then q (a 1, a 2 ) = 1 (x 1 x 2 + y 1 y 2 + z 1 z 2 ) 2 (x y z2 1 ) (x y z2 2 ). The projective spread S (L 1, L 2 ) is the spread between the projective lines, or their normals. If L 1 = (l 1 : m 1 : m 1 ) and L 2 = (l 2 : m 2 : n 2 ) then S (L 1, L 2 ) = 1 (l 1 l 2 + m 1 m 2 + n 1 n 2 ) 2 (l m2 1 + n2 1 ) (l m2 2 + n2 2 ).

17 A projective triangle Three projective points a 1, a 2, a 3 form a projective triangle, or tripod.the associated projective lines are L 1 = a 2 a 3, L 2 = a 1 a 3 and L 3 = a 1 a 2. Define the projective quadrances q 1 = q (a 2, a 3 ) q 2 = q (a 1, a 3 ) q 3 = q (a 1, a 2 ) and the projective spreads S 1 = S (L 2, L 3 ) S 2 = S (L 1, L 3 ) S 3 = S (L 1, L 2 ).

18 Projective rational trigonometry It turns out that these laws hold also for (universal) hyperbolic geometry! Projective spread law Projective cross law Dual projective cross law S 1 q 1 = S 2 q 2 = S 3 q 3. (S 3 q 1 q 2 q 1 q 2 q 3 + 2) 2 = 4 (1 q 1 ) (1 q 2 ) (1 q 3 ). (S 1 S 2 q 3 S 1 S 2 S 3 + 2) 2 = 4 (1 S 1 ) (1 S 2 ) (1 S 3 ).

19 Projective quadrea S 1 q 1 = S 2 q 2 = S 3 q 3 S q 1 q 2 q 3 Projective quadrea: S = S 1 q 2 q 3 = S 2 q 1 q 3 = S 3 q 1 q 2 Using altitude quadrances h 1, h 2, h 3, the Spread law S 1 = h 2 /q 3 gives S = h 1 q 1 = h 2 q 2 = h 3 q 3

20 (There is also a more advanced system emplying green struts). N J Wildberger () Tetrahedron trigonometry July / 36 The ZOME construction system The ZOME system was invented by Steve Baer and his wife Holly in 1992, with design work by Marc Pelletier and Paul Hildebrandt. Struts of three different colours (blue, red, yellow), and three different sizes, fit into white balls with 62 holes. The system allows construction of the Platonic solids and many related objects, including 3D models of the 120 and 600 cells.

21 ZOME quadrances By constructing a pentagon and some right triangles, we can determine the quadrances of the ZOME struts to be where τ = B 1 = 1 B 2 = τ B 3 = τ 2 R 1 = β R 2 = βτ R 3 = βτ 2 Y 1 = 3/4 Y 2 = 3τ/4 Y 3 = 3τ 2 /4. = ( 1 + ) and β = N J Wildberger () Tetrahedron trigonometry July / 36

22 ZOME and spreads The following spreads figure prominently in the regular pentagon, and are the zeroes of the 5-th spread polynomial S 5 (s) = s ( 5 20s + 16s 2) : α = β = Note that α 36, 144 and β 72, 108. With ZOME, we can easily construct an equilateral triangle (with equal spreads s = 3/4) and the sharp and flat triangles.

23 Two coloured primitive ZOME triangles τ = and τ = 3 5 2

24 Three coloured primitive ZOME triangles (This is not quite a complete list of ZOME triangles: there are also a few more that require more than one strut per side).

25 A ZOME tetrahedra Here is one of the 65 possible ZOME tetrahedron. It arises naturally when we build simultaneously both a dodecahedron and an icosahedron with the same center, using ZOME. Here are the planar quadrances and spreads of this tetrahedron, formed by the four triangular faces.

26 Tartaglia s volume formula (also Euler, Cayley-Menger) v 2 = det 2Q 12 Q 12 + Q 13 Q 23 Q 12 + Q 14 Q 24 Q 12 + Q 13 Q 23 2Q 13 Q 13 + Q 14 Q 34 Q 12 + Q 14 Q 24 Q 13 + Q 14 Q 34 2Q 14 Tartaglia s formula The quadrume of the tetrahedron V = 144v 2 is given by V = Q 12 Q 13 Q 24 + Q 12 Q 13 Q 34 + Q 12 Q 14 Q 23 + Q 12 Q 14 Q 34 + Q 12 Q 23 Q 34 + Q 13 Q 24 Q 34 + Q 12 Q 24 Q 34 + Q 14 Q 23 Q 24 + Q 14 Q 23 Q 34 + Q 13 Q 14 Q 23 + Q 13 Q 14 Q 24 + Q 13 Q 23 Q 24 Q 12 Q 14 Q 24 Q 23 Q 24 Q 34 Q 12 Q 13 Q 23 Q 13 Q 14 Q 34 Q 13 Q 2 24 Q 12Q 2 34 Q2 13 Q 24 Q 2 12 Q 34 Q 2 14 Q 23 Q 14 Q 2 23.

27 The Alternate spreads theorem Relations between 12 planar face spreads s i,j : Four Triple spread formulas, one for each face, eg. (s 1,4 + s 2,4 + s 3,4 ) 2 = 2 ( s 2 1,4 + s 2 2,4 + s 2 3,4) + 4s1,4 s 2,4 s 3,4 Alternate spreads theorem s 2,4 s 3,4 s 3,2 s 4,2 s 4,3 s 2,3 = 1. There are four such relations, and the product of them is identically 1 = 1.

28 The trigonometry of a tetrahedron In a tetrahedron, the Projective cross law has a lovely reformulation that allows us to express the dihedral edge spreads E ij in terms of Q kl. (E 12 s 1,3 s 1,4 s 1,2 s 1,3 s 1,4 + 2) 2 = 4 (1 s 1,2 ) (1 s 1,3 ) (1 s 1,4 ) (E 12 s 1,3 s 1,4 s 1,2 s 1,3 s 1,4 + 2) 2 = 4 (Q 34 Q 13 Q 14 ) 2 (Q 24 Q 12 Q 14 ) 2 (Q 23 Q 12 Q 13 ) 2 4Q 13 Q 14 4Q 12 Q 14 4Q 12 Q 13 = (Q 34 Q 13 Q 14 ) 2 (Q 24 Q 12 Q 14 ) 2 (Q 23 Q 12 Q 13 ) 2 16Q12 2. Q2 13 Q2 14

29 Dihedral spreads and quadrances Recall quadrea A = (Q 1 + Q 2 + Q 3 ) 2 2 ( Q Q2 2 + Q2 3 ) A (Q1, Q 2, Q 3 ). Dihedral spreads theorem By symmetry we have E 12 = 4Q 12V A 3 A 4. from which we get V = E 12A 3 A 4 4Q 12 = E 13A 2 A 4 4Q 13 = E 14A 2 A 3 4Q 14 = E 23A 1 A 4 4Q 23 = E 24A 1 A 3 4Q 24 = E 34A 1 A 2 4Q 34 E 12 E 34 Q 12 Q 34 = E 13E 24 Q 13 Q 24 = E 14E 23 Q 14 Q 23 = 16V 2 A 1 A 2 A 3 A 4.

30 Solid spreads The four projective quadreas S 1, S 2, S 3, S 4 are the solid spreads of the tetrahedron. Recall S 1 = E 12 s 1,3 s 1,4 = E 13 s 1,2 s 1,4 = E 14 s 1,2 s 1,3. Solid spread theorem S 1 = V 4Q 12 Q 13 Q 14. Proof: Using the Dihedral spreads theorem, S 1 = ( ) 4Q12 V E 12 s 1,3 s 1,4 = s 1,3 s 1,4 A 3 A 4 = 4Q 12 V (4Q 12 Q 14 s 1,3 ) (4Q 12 Q 13 s 1,4 ) s V 1,3s 1,4 =. 4Q 12 Q 13 Q 14 As Corollaries: S 1 = Q 23Q 24 S 1 S 2 = Q2 34. S 2 Q 13 Q 14 S 3 S 4 Q 2 12

31 Solid spread law As another corollary of S 1 = Solid spread law V 4Q 12 Q 13 Q 14 we get S 1 Q 23 Q 24 Q 34 = = S 2 Q 13 Q 14 Q 34 = S 3 Q 12 Q 14 Q 24 = V 4Q 12 Q 13 Q 14 Q 23 Q 24 Q 34. S 4 Q 12 Q 13 Q 23 In addition to E 12 = 4Q 12V A 3 A 4 we have also the shorter result = 1 E 12 ( Q Q ) 2 12Q 34 + Q 13 Q 14 + Q 23 Q 24 Q 12 Q 13 Q 12 Q 14 Q 12 Q 23 Q 12 Q 24 Q 13 Q 24 Q 14 Q 23. A 3 A 4

32 The right tetrahedron The special case of a right tetrahedron, where say S 1 = 1, deserves attention, since it is the three dimensional analog of a right triangle. Right Pythagoras theorem If the tetrahedron has solid spread S 1 = 1 then A 1 = A 2 + A 3 + A 4. Right solid spreads If the tetrahedron has solid spread S 1 = 1 then (S 2 + S 3 + S 4 1) 2 = 4S 2 S 3 S 4. This relation is precisely the Triple cross formula, i.e. the quantities C 1 = 1 S 1, C 2 = 1 S 2 and C 3 = 1 S 3 satisfy the Triple spread formula!

33 Dihedral spreads relation (complicated) Dihedral spread relation If a 1 E 12 b 1 E 13 c 1 E 14 and x 1 E 34 y 1 E 24 z 1 E 23 and t 1 2 (E 12E 34 + E 13 E 24 + E 14 E 23 ) 1 f 1 2 (E 12E 34 + E 13 E 24 E 14 E 23 ) g 1 2 (E 12E 34 E 13 E 24 + E 14 E 23 ) h 1 2 ( E 12E 34 + E 13 E 24 + E 14 E 23 ) u t2 + abxy + bcyz + acxz (abc + ayz + bxz + cxy) 2 v u2 ( abxyf 2 + acxzg 2 + bcyzh 2) 2 then v 2 = abcxyz ( af 2 g 2 x + bf 2 h 2 y + cg 2 h 2 z + 2fghu ).

34 Dihedral spreads formula Suppose that Q 12 = 1 Q 13 = 2 Q 14 = 3 Q 23 = 2 Q 24 = 5 Q 34 = 4 Then E 12 = E 13 = E 14 = E 23 = E 24 = E If E 12 was unknown, it would satisfy the polynomial relation 0 = (77E 68) (28E 17) ( E E ) ( 5929E E ) ( 5929E E ). So perhaps there are "linked tetrahedra" that share 5 dihedral spreads?

35 Our example tetrahedron Q 12 = βτ Q 13 = 3τ/4 Q 14 = 3τ/4 Q 23 = Q 24 = β Q 34 = 1 V = A 1 = A2 = A 3 = A 4 = E 12 = E 23 = E 24 = β E 13 = E 14 = 3 4 E 34 = 1 S 1 = S2 = S3 = S 4 =

36 Circumdiameter quadrance Theorem (Crelle 1821) The circumdiameter quadrance D is D = A V where A = A (Q 12 Q 34, Q 13 Q 24, Q 14 Q 23 ). Example In our tetrahedron, D = A V = ( ) 5 /V =

37 Some ZOME challenges (Picture thanks to Alejandra Plaice). The formulas in this talk are suffi cient to study the 65 possible ZOME tetrahedra. (1) 1 2 Each can now be analysed metrically. A challenge: analyse metrically every ZOME tetrahedron. 3 2.png A harder theoretical challenge: what further relations are there between these various quantities (and others)? What are the relations between the relations? What are the possibilities when we extend ZOME to the green struts?!

38 Two key examples Here are the two essential triangles that are studied by all students, essentially the only ones that are accessible to elementary classical trigonometric, but now expressed using rational trigonometry. Note the absence of any transcendental quantities these are fictions induced by a myopic and distorted view. What are the three dimensional analogs??

39 The right isosceles tetrahedron Suppose that we have a right tetrahedron with Q 12 = Q 13 = Q 14 = 1 while Q 23 = Q 24 = Q 34 = 2. Then the quadrume is V = 4. The quadreas are A 1 = 12 and A 2 = A 3 = A 4 = 4 The edge dihedral spreads are The solid spreads are E 12 = E 13 = E 14 = 1 and E 23 = E 34 = E 24 = 2 3. S 1 = 1 and S 2 = S 3 = S 4 = 1 4. Note that (S 2 + S 3 + S 4 1) 2 = 4S 2 S 3 S 4.

40 The regular tetrahedron Suppose that all quadrances Q ij = 1. Then the quadreas of the faces are all A = 3 and the quadrume is The faces spreads are all the edge spreads are all and the solid spreads are all V = 2. s ij = 3/4 E ij = 8/9 S i = 1 2. Rational trigonometry allows us to metrically describe the regular tetrahedron in a way that I hope Pythagoras and Euclid would approve!

41 More info BOOK: N J Wildberger, Divine Proportions: Rational Trigonometry to Universal Geometry (WildEgg Books 2005) ARTICLES: Universal Hyperbolic Geometry I: Trigonometry Geometria Dedicata (2012) Universal Hyperbolic Geometry II: A pictorial overview KoG (2010) Universal Hyperbolic Geometry III: First steps in projective triangle geometry KoG (2011) YOUTUBE VIDEOS (njwildberger): WildTrig (Rational Trigonometry) UnivHypGeom (Universal Hyperbolic Geometry) Hyperbolic geometry is projective relativistic geometry Triangle geometry old and new: An introduction to hyperbolic triangle geometry THANK YOU! N J Wildberger () Tetrahedron trigonometry July / 36