An Asymptotically Closed Loop of Tetrahedra

Transcription

1 Nov 06; to appear, Mathematical Intelligencer An Asymptotically Closed Loop of Tetrahedra Michael Elgersma and Stan Wagon Introduction In 958 S. Świerczkowski [5], answering a question of Steinhaus, proved that there cannot be a closed loop of regular tetrahedra meeting face-toface (even ignoring the issue of whether the chain is embedded no self-intersection). Such embedded loops do exist for the other four Platonic solids (see []). A natural question is how close a chain can come to closure and in [] we conjectured that the discrepancy from closure can be arbitrarily small. Here we present a simple pattern that settles this question affirmatively. A more complete discussion is in []. Here is how Świerczkowski phrased the question in his unpublished memoir [6]. Granted then, that the last pyramid in a Steinhaus chain never can have a sidewall in common with the first pyramid P, it still may happen that all observations and measurements indicate that these two pyramids do have a sidewall in common. This would not contradict the mathematical result; it would only illustrate the obvious fact that no measurement is 00% accurate. So, a new problem is born: Whatever threshold of accuracy is selected, say, represented by a (small) number ϵ, will there be a chain of pyramids, returning to P such that within the accuracy of ϵ inches, the last pyramid of the chain has indeed a sidewall in common with P. It is hard to tell if anyone will ever want to devote her or his time to search for an answer to this question. In any case, it is unlikely that an answer would be easily found. Our main theorem finds the answer. It is not surprising that nearly closed chains do exist, but it was pleasing to find that there is a simple pattern that proves it. The proof combines geometry and number theory, as continued fractions play a role. Theorem. For any ϵ > 0, there is an embedded chain of regular tetrahedra, meeting face-to-face, so that the discrepancy from a perfect loop is less than ϵ. We need some precise definitions. Definition A tetrahedral chain is a finite sequence of congruent regular tetrahedra meeting face to face, but never doubling back through a justused face. The gap of a chain is the smallest d such that one can move each of the vertices of an exterior triangular face of the chain a distance no greater than d so that the chain becomes a closed loop. The Tetrahelix The tetrahelix (also known as the Boerdijk Coxeter helix) is our basic tool: it s a stack of tetrahedra corresponding to reflections in the faces,,, 4,,,, 4,,,, 4,. A simple explicit representation [4] can be given using cylindrical coordinates: let θ = cos - ( /), r = 0, h = 0, and V i = (r cos(i θ), r sin(i θ), i h); then the tetrahelix TH L consists of the L tetrahedra formed by V 0, V,, V L+ (Fig. ). For these points the tetrahedral edges have unit length. Note that there are left-handed and right-handed tetrahelices: the canonical TH L using the coordinates just given is right-handed as it rises in a counterclockwise way as viewed from above (or: it rises around its axis so as to obey the right-hand rule); a reflection in any plane would change it to a left-handed tetrahelix. Figure. The tetrahelix made from 6 tetrahedra colored, in order, red, green, blue, yellow, red, green, blue, yellow, and so on. The vertices are equally spaced along a helix.

2 The Quadrahelix The quadrahelix QHL is made from four tetrahelices as follows. Start with a pivot tetrahedron (gray in Fig. ) and attach a right-handed copy of THL onto face (red in Fig. ) and a left-handed THL onto face (green in Fig. ). This shape is symmetric in the reflection plane we call this the first quadplane that splits the pivot and sends face to face. Then reflect (as in Fig. (b)) this V-shaped object in the plane we call this the biplane defined by a terminal face of one of the tetrahelices (we always use the terminal green face as in Figs. (b) and ). The second quadplane is the reflection of the first in the biplane. (a) (b) Figure. (a) The first half of the quadrahelix QH,with the first quadplane. (b) The complete quadrahelix Q. Each QHL has four legs and the angles between the tetrahelical axes work out to be, in order, sec- (-5), sec- (5), and sec- (-5). But the exciting thing about this shape is that, for special L-values, it comes very close to a perfect loop. Figure shows how this works nicely when L = 0 (see also Fig. 7). It turns out that a small gap corresponds to a small value of the reduced angle (L + ) θ. Let δ be the mod- π reduction of (L + ) θ, with -π δ < π. Figure. QH0 has a gap that is about / the side length. Three-dimensional printing can be used to get a physical model; Figure 4 shows a Shapeways model (available at [7]) of QH0, with the gap in black. The quadrahelix pattern is so simple that it would seem not difficult to use a polyhedron construction tool such as ZomeTool or Polydron to transform QH0 or QH9 into a hoax: a mathematically impossible closed loop. Figure 4. A colored sandstone model of QH0, printed by Shapeways.

3 One can give a simple geometric explanation of why a small δ leads to a nearly closed loop. Let ϕ (with 0 < ϕ < π) be the dihedral angle between the first quadplane and the start plane (the plane determined by the floor of T ; see Fig. 5); let ρ = π - ϕ; and assume QH L consists of tetrahedra T, T,, T n starting from the red tetrahedron farthest from the first pivot. Imagine starting at T and going one tetrahelical step farther than the pivot T L+. That corresponds to L + rotations by θ. Therefore having δ close to 0 means that the total angular change around the tetrahelix axis is small and the resulting tetrahedron is close to a translation of T. So the ceiling of the pivot tetrahedron is almost parallel to the floor of T and the quadplane is almost orthogonal to the start plane; equivalently, ρ is small. The last face of the green leg is therefore both a reflection of and almost parallel to the first face of T and the reflection of the first half that completes the quadrahelix has a terminal face that is almost identical to the initial face. This means that the loop is almost closed. To turn this into a rigorous proof we will derive this exact relation between δ and ρ: sin ρ = 5 important. 6 sin δ. This identity shows clearly that small δ implies very small ρ; the quadratic nature of the relation turns out to be critically Figure 5. The quadrahelix has three symmetry planes. These planes, and the planes defined by the two ends, all pass through a common rotation axis. Shown here is QH 5 ; the two orange points at top are V L+ and V L+ ; the vector they determine is normal to the first quadplane. Embedding Proof The proof that the quadrahelix is embedded relies on the facts that the five planes in Figure 5 divide space into four regions and the planes meet in an acute angle. Let C = 5 6. Five-Plane Lemma. The biplane, first quadplane, second quadplane, start plane, and end plane all pass through a common line, called the rotation axis. Proof. The start plane meets the first quadplane in a line; the start plane and the biplane are reflections in the first quadplane; therefore the biplane meets the first quadplane in the same line. Reflection in the biplane yields the remaining collinearities. See Figure 5. 4 Acute Angle Lemma (a). The angle between A, the vector normal to the tetrahelix's base triangle V 0 V V and pointing in the direction of V, and the vector V L+ - V L+ is acute. (b). Angle ϕ is acute, cos - C ϕ < π, and 0 < ρ sin- C. (c). cos ϕ = sin ρ = C sin δ. Proof. By direct computation: A = 6 trig expansion, ( V - (V 0 + V + V )) = 5 0,, and, using the substitution L θ δ - θ and some V L+ - V L+ = cos δ - sin δ, cos δ + 5 sin δ, 6. It follows that A (V L+ - V L+ ) = C ( - cos δ) = C sin δ. If δ = 0, then the dot product vanishes, the angle in question is exactly 90 and the

4 loop closes up perfectly, contradicting Świerczkowski's Theorem. Therefore the dot product satisfies C A V L+ V L+ 0 and the angle is acute and greater than cos - C (about ). Because the first quadplane is defined by the bisected pivot tetrahedron, its normal vector is the edge of that tetrahedron connecting the tetrahelix vertices V L+ and V L+ (the orange points in Fig. 5). Therefore ϕ is exactly the angle in (a); (b) and (c) then follow immediately from (a) and its proof. 4 Theorem. For any L, QH L is embedded. Proof. A tetrahelix has no collision, so each colored sector is embedded. The next step is showing that the first sector (red) stays within the region defined by the start plane and the first quadplane. The symmetry of construction then yields the same for the other colors and the appropriate planes. For this proof, we can view the pivot tetrahedron attached to the red tetrahelix as being the start; then the three points defining the quadplane are V, V, and (V 0 + V ). The sign of D, the determinant of the matrix V q - V, V q - V, V q - (V 0 + V ), determines which side of the plane contains V q. Letting c = cos (q θ) and s = sin(q θ), the matrix is this: 4 c + s - 5 (q - ) h r r c + 9 s (q - ) h r c s q - h r and the value of 0 0 D works out to 6 5 q c + 7 s, which is not less than q - 0. Because q in this approach, D is positive, as required. A similar and easier computation shows that the red tetrahelix stays on the proper side of the start plane. The Five-Plane and Acute Angle Lemmas mean that the five planes define four regions containing the four colored sectors of the quadrahelix and there are no collisions between colors. In particular, the total angle as one moves around the rotation axis is 4ϕ < 60 and the tetrahedra at the ends of the chain are disjoint. 4 The Vanishing Gap Recall that δ is the reduction of (L + ) θ modulo π. If δ were 0, then, by Corollary, ϕ would be exactly 90 and QH L would close up perfectly: the four right angles would combine to make a perfect loop. This cannot happen (by either Świerczkowski s Thm. or the irrationality of θ / π), but we can try to make δ small. The convergents k q close to the integer k. More precisely, the error θ π - k q is bounded by that >δ? < π 5 (L+) θ (L+) θ of the continued fraction of give arbitrarily large values q = L + for which is very π π 5 q (Hurwitz's Thm.; [, Exer. 7.0]). Multiplication by π q implies. This bound will be used in the next proof to show that the gap can be made arbitrarily small. Theorem 5. For any ϵ > 0, there is a quadrahelix QH L having gap less than ϵ. Proof. Using the continued fraction method just discussed, choose L so that >δ? < π 5 (L+) and L > 8 ; the first condition and part (c) of the ϵ Acute Angle Lemma imply that ρ < 5. This means that the rotation axis lies inside the quadrahelix (by which we meaning that the plane orthogonal to the biplane and containing the rotation axis intersects the quadrahelix), since otherwise ϕ < π and ρ > π 4.

5 7+ 5 Figure 6. When viewed as a projection perpendicular to the rotation axis, the gap bound is a leg of a right triangle with hypotenuse H. Consider the projection along the rotation axis. Because the axis lies inside the quadrahelix, it follows that the gap is bounded by twice the length of the leg of a right triangle whose hypotenuse joins the rotation axis and the farthest point in the triangle defining the start plane and with the right angle formed by the perpendicular from this line to the biplane (Fig. 6). The triangle s angle at the rotation axis is ( π - 4 ϕ) = ρ and a rough estimate shows that the hypotenuse length H is less than ( L + ) h. Therefore, using part (c) of the Acute Angle Lemma to eliminate ρ and the relation between δ and L to eliminate δ. gap H sin (ρ) 4 ( L + ) h sin ρ = 4 L h C sin δ L 5 5 δ = δ L < π 5 L L 8 L < ϵ 4 Note that in the preceding proof it is critical that we have the quadratic expression q - in the bound from the continued fraction approximation. This is because we have to absorb the hypotenuse length, which is O(L). The continued fraction of θ π is and so the convergents are,, 8, 4, 0, 5 4, 6 7, 67 8, 9 54, 78 96, , 59 98, , , Computing these out to the millionth term is very fast. So we learn that L =, 7, 0, 9, 40, 70, yield chains that approach perfect closure; see Table. And for the following 99-digit integer, L = then the gap size of QH L is less than One can even get a value of L having digits so that the gap size of QH L is less than 0 -,000,000. Figure 7 shows QH 70 ; the limiting shape as the gap approaches is 0 is a rhombus.

6 6 L log 0 >δ? upper bound on log 0 (gap) Table. Values of L for which QH L is almost closed. The gaps are roughly inversely proportional to L. Editor: Are black backgrounds better for the clean unlabeled images, like Figs and 7? I think so, and have used Black in this draft. And these clean images are likely best if you are considering a cover. At L=9 and 40 the gap is visible, so the hoax effect is ruined. Of course, I can adjust these images any way you like, or make a composite image out of four of these chains. Figure 7. QH 70 has a gap that is invisible at this scale. Conclusion While the quadrahelix appears to be relatively simple, finding it was anything but. Experiments related to our work in [] led us to an 8-sided shape that did serve to prove the theorem, but was a little complicated. Then we more or less stumbled onto the 4-sided shape QH L that works, has a simpler proof, and might be the simplest possible pattern that resolves the conjecture. Among unresolved issues, one question stands out. The quadrahelix requires O( / ϵ) tetrahedra to achieve a gap of ϵ; it takes more than 0 6 tetrahedra to get a quadrahelical gap near 0-7. Our earlier searches as in [] led to an embedded chain of 540 tetrahedra with a gap of (Fig. 8). Experiments [, Fig. 9] suggest that a gap of ϵ can be achieved with 0 log( / ϵ) tetrahedra, a formula that predicts about 400 tetrahedra for ϵ = 0-7. So we ask: Is there a pattern for embedded chains that achieves gap ϵ using O(log( / ϵ)) tetrahedra?

7 7 Figure 8. An embedded chain of 540 tetrahedra that has a gap smaller than 0-7. Acknowledgement. Our discoveries and proofs would not have been possible without the comprehensive graphic and algebraic capabilities of Mathematica. References. D. Bressoud and S. Wagon, A Course in Computational Number Theory, New York, Wiley, M. Elgersma and S. Wagon, Closing a Platonic gap, The Mathematical Intelligencer 7: (05) , The quadrahelix: A nearly perfect loop of tetrahedra, Colloquium Mathematicum (submitted), 4. R. W. Gray, Tetrahelix data ; accessed Nov., S. Świerczkowski, On chains of regular tetrahedra, Coll. Math. 7 (959) , Looking Astern, unpublished memoir. 7. S. Wagon, Quadrahelix0, Shapeways