Journal for Geometry and Graphics Volume 6 (2002), No. 2, 167{182. Golden Hexagons Gunter Wei¼ Institute for Geometry, Dresden University of Technology Zellescher Weg 12-14, Willersbau B 119, D-01062 Dresden, Germany email: [email protected] Dedicated to Prof. Hellmuth Stachel on the occasion of his 60th birthday Abstract. A \golden hexagon" is a set of six points, which is projectively equiva- lent to the vertices of a regular pentagon together with its center. Such a geometric ¯gure generalizes in some sense the classical one-dimensional golden section to two dimensions. This paper deals with some remarkable properties of golden hexagons and with special Euclidean representatives as well as with further generalizations. Keywords: golden section, golden ratio, golden cross ratio, geometrically de¯ned iterative processes, Desargues' Theorem, polarity with respect to a conic, Moe- bius circle geometry, bio-geometry, regular polyhedra MSC 2000: 51M04, 51M05, 51M20 1. An iterative process based on the polarity with respect to a conic In the real projective plane ¼ let a set of six arbitrarily chosen points P be given such that P j each selected quintuple P , i = 1; : : : ; 6, de¯nes a regular conic c , and not all six points P n f ig i belong to the same conic. We proceed as follows: In a ¯rst step we polarize P at c for i = 1; : : : ; 6. This gives a set of six lines ² i i L l1; : : : ; l6 in ¼. In general, each quintuple of these lines again de¯nes a (dual) conic and its polarity. Another polarizing step | dual to the ¯rst step | produces again a hexagon 0. ² P Obviously, step one and its dual start an iteration process 0 00 : : : P 7! P 7! P 7! in ¼ which is connected to the ¯rst hexagon in a projectively invariant way. P There the question arises: To what extent does an eventually existing limit hexagon 1 P depend on the initial ¯gure , and do there exist additional characterizing properties of 1 ? P P ISSN 1433-8157/$ 2.50 c 2002 Heldermann Verlag ° 168 G. Wei¼: Golden Hexagons An analytical treatment can be based on projective coordinates [5], [4] (i) (i) (i) (i) (i) (i) R(x0 ; x1 ; x2 ) of Pi ¼; (y0 ; y1 ; y2 )R of li ¼; i = 1; : : : ; 6: (1) 2 ½ E.g., the equation (6) 2 (6) 2 (6) 2 (6) (6) (6) a00 x0 + a11 x1 + a22 x2 + 2a01 x0x1 + 2a02 x0x2 + 2a12 x1x2 = 0 (2) (6) describes the conic c6 de¯ned by the ¯rst ¯ve points Pi, whereby the coe±cients ajk of (2) result from the system of linear equations 2 2 2 x x0 x1 x2 x0x1 x0x2 x1x2 (1) 2 (1) (1) (1) (1) P1 (x0 ) : : : x0 x0 : : : x1 x2 0 (2) 2 (2) (2) P2 (x0 ) x1 x2 0 . (3) . (5) 2 (5) (5) P5 (x0 ) x1 x2 0 (6) (6) (6) (6) (6) (6) a00 a11 a22 a01 a02 a12 The coe±cients a(6) are the six main minors of the 6 5-matrix (3). jk £ Finally, the polar line l6 to point P6, with respect to the conic c6 has (dual) projective coordinates (6) (6) (6) (6) a00 a01 a02 x0 (6) (6) (6) 0 (6) (6) (6) 1 0 (6) 1 (y0 ; y1 ; y2 ) = a01 a11 a12 x1 : (4) (6) (6) (6) ¢ (6) B a a a C B x C @ 02 12 22 A @ 2 A (i) Thus, the ¯rst step ends with line-coordinates yk which are forms of degree 11 in the coordinates of the given points P1; : : : ; P6. Therefore, by the analogous dual step, the co- 0 ordinates of a point Pi are forms of degree 121 in the coordinates of the initial points Pj. Consequently, there hardly seems to be any chance to describe the iteration process and its limit ¯gure 1 by means of computer aided algebraic manipulations. P 1.1. Some experiments Numerical treatment shows that for any (general) initial hexagon there exists a \limit" ¯gure 1 (see Fig. 1). We will show that all the limiting ¯gures of di®erenP t initial ¯gures are P projectively equivalent (see Section 2). Obviously, a hexagon consisting of six points of one conic, | we call such a hexagon P Pascal-hexagon, as the six points obey Pascal's condition (c.f. [5]) | should be ¯xed under the iterated polarizing process IPP described above. Pascal-hexagons turn out to be repulsive initial ¯gures (Fig. 2). An initial hexagon with (Euclidean) symmetries will lead to derived hexagons 0, 00, P P P . with the same symmetries. Especially a regular pentagon together with its center is ¯xed under IPP. This can be shown by elementary geometric arguments (Fig. 3). Another special Euclidean representative of an attractive ¯xed hexagon with a three- Ptri fold symmetry consists of two concentric equilateral triangles with a side length ratio of s : s = 1 : '4; (' = 0; 618:::): (5) 1 2 ¡ Fig. 4 shows remarkable incidences of sides of : Ptri G. Wei¼: Golden Hexagons 169 PSfrag replacements PSfrag replacements P L PSfrag replacements PSfrag replacemen0 ts 0 P L PSfrag replacements PSfrag replacemen00 ts 00 P L Figure 1: First six steps of an iterated polarizing process IPP applied to the initial hexagon P The point A of intersection between any side of the \inner" triangle and a side of the ² \outer" one divides the outer (and the inner) side according to the golden section. Connecting A with any point of gives a line containing two points of . ² Ptri Ptri So, in spite of the fact that collinear transformations in ¼ do not preserve ratios (only cross- ratios), a perspective collineation transforming into pent (see Fig. 5) seemingly preserves Ptri P the \golden proportions". So it seems worthwhile to take a closer look to golden proportions from the projective point of view (see Section 3). Because of their golden proportions we will call , and all their projectively equivalent hexagons 1 \golden hexagons" . Ppent Ptri P Pgold 170 G. Wei¼: Golden Hexagons PSfrag replacements PSfrag Preplacements L 0 0 PSfrag replacements PSfrag Preplacements L 00 00 PSfrag replacements PSfrag Preplacements L Figure 2: The iterated polarizing process IPP applied to an initial hexagon , which is almost a Pascal-hexagon P 2. Projective geometric properties of golden hexagons One might \standardize" the iterative polarizing process IPP by adding a collinear transfor- 0 0 mation · to each pair of dual steps, which maps points P1; : : : ; P4 onto P1; : : : ; P4, respectively. This means that the \extended IPP" keeps the ¯rst four points of ¯xed. In the following P these points will serve as the projective coordinate frame in ¼. So the question arises: To 1 1 which limit points P5 , P6 will an arbitrarily chosen pair of points P5; P6 tend? Fig. 6 shows that, according to the initial position of the pair (P5; P6), there are six pos- sibilities (P 1; P 1) completing P ; : : : ; P to a limit hexagon 1, and we learn from the 5 6 f 1 4g P coordinates of (P 1; P 1) that 1 is always a golden hexagon. Experiments show that ac- 5 6 P G. Wei¼: Golden Hexagons 171 T6 T 3 T5 4 t5 t3 T1 T2 5 3 6 t6 4* A D B C 2* 1* 1 2 t1 t2 5* 3* Figure 3: A regular pentagon and its center as an attractive ¯xed hexagon pent P under the iterative polarizing process IPP, together with ¤ the conic c4 and its polar parabola c4 5 B 4 6 . 2 . 1 3 Figure 4: The attractive ¯xed hexagon and its golden proportions Ptri 172 G. Wei¼: Golden Hexagons B 1′ B′ O 1 A′ 2′ 6′ 4′ 3 ′ 5 ′ a 3 5 4 A 2 6 Figure 5: Perspective collinear transformation of into Ppent Ptri (center O and axis a) ∞ P5 P3 P2 ∞ P6 P4 P1 Figure 6: Limit ¯gures of an extended iterative polarizing process IPP Pgold applied to a hexagon with four points of remaining ¯xed at all steps of IPP P P cording to the initial position of the randomly chosen pair (P5; P6) there occur six possibilities 1 1 for the pair of limit points (P5 ; P6 ), each point having \golden" coordinates. Combinatorial arguments for choosing start positions support the 1 1 Conjecture that there exist exactly these six pairs of limit points and that P1; : : : ; P4; P5 ; P6 in any case form a golden hexagon. G. Wei¼: Golden Hexagons 173 In Fig. 6 the pairs of limit points are marked by the same symbol and one read o® their 1 1 coordinates that the set of all possibilities P5 resp. P6 each forms a golden hexagon, too. These new golden hexagons can be based on a square as follows: Let three vertices of a square be the ¯rst three points of a golden hexagon gold. Divide the sides having the fourth vertex in common in accordance to the golden ratio,Pthe \minor" closer to that fourth vertex, and receive so the 4th and 5th point of . Connect these Pgold points by a line and reflect the fourth vertex of the square at this line receiving ¯nally the 6th point of . Note that this special Euclidean representative of again shows golden Pgold Pgold ratios in spite of its projective origin! As a main result we state: Theorem 1 Each iterative polarizing process IPP has a golden hexagon as attractor 1, P which is projectively equivalent to pent. P Golden hexagons have another remarkable property, which is characteristic with respect to the group of projective transformations: Theorem 2 Each partition of a golden hexagon into two triangles allows a homology from one triangle to the other, i.e., all pairs of triangles are in \Desarguesian position".