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Additional info for The Mathematical Intelligencer Vol. 32, No. 2
After consulting some colleagues as to whether the problem was known, and following a great deal of experimentation, Hill found drawings of K6 with 3 crossings, K7 with 9 crossings, K8 with 18 crossings, and K9 with 36 crossings. Figure 10 shows some of his geometrical jottings. After a great deal of experimentation, Hill also produced a construction that can be described as follows: Figure 8. Anthony Hill with geometrical objects. Anthony Hill Anthony Hill (b. 1930) describes himself as a ‘constructivist working as a geometric formalist’ and has been described by the architect Yona Friedman (see [15, p.
It yields the following polynomial with three free parameters: f ðu; v; wÞ ¼ 1 À 1 þ c1 c4 u þ c1 u2 þ c2 uv À 6v À 3v 2 þ 8w; c2 ð29Þ with -(1 + c1c4) + c2c2 \ 0. Note that the secondary invariant s2 does not appear in the above polynomial, its coefficient b is 0. We obtain a nice result for c1 = c2 = 0, c = 1/3; see Figure 14a. E X A M P L E 12 (D4). The group D4 has the following primary and secondary invariants: uðx; y; zÞ ¼ z 2 ; vðx; y; zÞ ¼ x 2 þ y 2 ; Its secondary invariant is s2 ðx; y; zÞ ¼ 5x 4 yz À 10x 2 y3 z þ y5 z: ð34Þ A degree five polynomial already produces a solvable system of equations, but the resulting polynomial with three free parameters only permits anti-stars.
The hypocycloid with four cusps is also called the Astroid. Its implicit equation is (1 - u)3 - 27v = 0. So if we choose c1 = -27 and c2 = 0 in (23) we obtain the same curve. See Figure 12c. E X A M P L E 10 (5-star). The primary invariants of D5 are uðx; yÞ ¼ x 2 þ y 2 ; vðx; yÞ ¼ x 5 À 10x 3 y2 þ 5xy4 : ð24Þ If we try a degree five polynomial, we obtain (25) with no free parameters. It only permits anti-stars. f ðu; vÞ ¼ 1 À 10 8 u þ 5u2 À v: 3 3 type A1, that is, they have, up to analytic coordinate transformations, equation x2 + y2 = 0.
The Mathematical Intelligencer Vol. 32, No. 2 by Chandler Davis (editors-in-chief)