The following table updated from Friedman , and with the edge Mireles graph, which is not rigid, removed gives smallest known solutions as of Oct. Khodulyov's solutions for the heptagon and gon are just instances of an "equal angles" method using Peaucellier-Lipkin linkages that works for all -gons with with the exceptions of , 9, 10, and 12, which produce vertex-vertex degenerate embeddings. The result is asymptotically optimal and requires rods J. Tan, pers. Frederickson, G. New York: Cambridge University Press, pp.
Friedman, E. Updated Oct. Gardner, M. Maehara, H. Pegg, E. Somsky, W. Taxel, P. Wells, D. London: Penguin, p. Weisstein, Eric W.
It can easily be transformed into a parallelogram. This is why you see triangles all over the place in the world around you. In electricity pylons, cranes, bridges, and many houses. Three-dimensional space is a little more permissive. Suppose you form a polyhedron by hinging together rigid faces at their edges. In the 19th century the French mathematician Augustin Louis Cauchy showed that all convex polyhedra are rigid. Convex means that any line connecting two points that are part of the polyhedron is also contained in the polyhedron.
This means that the five familiar Platonic solids are rigid. What can be said about non-convex polyhedra? Repeating this process builds a legitimate tiling of the plane. Thus, all triangles and quadrilaterals — even irregular ones — admit an edge-to-edge monohedral tiling of the plane. For example, consider the pentagon below, whose interior angles measure , , , and degrees. The pentagon above admits no monohedral, edge-to-edge tiling of the plane.
To prove this, we need only consider how multiple copies of this pentagon could possibly be arranged at a vertex. We know that at each vertex in our tiling the measures of the angles must sum to degrees. Constructing an irregular pentagon in this way shows us why not all irregular pentagons can tile the plane: There are certain restrictions on the angles that not all pentagons satisfy.
But even having a set of five angles that can form combinations that add up to degrees is not enough to guarantee that a given pentagon can tile the plane. Consider the pentagon below. This pentagon has been constructed to have angles of 90, 90, 90, and degrees. This means that when we attempt to create an edge-to-edge tiling of the plane, every side of this pentagon has only one possible match from another tile.
Knowing this, we can quickly determine that this pentagon admits no edge-to-edge tiling of the plane. Consider the side of length 1. Here are the only two possible ways of matching up two such pentagons on that side. The first creates a gap of 20 degrees, which can never be filled. The second creates a degree gap. We do have a degree angle to work with, but because of the edge restriction on the y side, we have only two options.
Neither of these arrangements generates valid edge-to-edge tilings. Thus, this particular pentagon cannot be used in an edge-to-edge tiling of the plane. We need five angles, each of which can combine with copies of itself and the others to sum to But we also need five sides that will fit together with those angles. But some just-right pentagons exist.
Things get trickier as we relax more conditions. When we remove the edge-to-edge restriction, we open up a whole new world of tilings.
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