Triangle-Closed Planar Sets

You are given a special triangle (or a set of three non-collinear points) as a seed for a subset of the plane.
For all pairs of points A, B in you are asked to find the "closure point" C = C() such that the triangle ABC is similar to .
Once C is found, we add it to the set .
is called the -closure or triangle-closure of .
The cases where is not a dense subset of the plane are the most interesting ones. Can you find a few?
Several examples of non-dense sets are given (choose one from the "closed set" options).


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Since rotation and size of the seed does not change the definition of -closed, a subset of the plane is called triangle-closed if it is (, )-closed for some real number and some angle . Then for each pair A, B of points in there is a point C in such that and .
In this game you define first (points A,B,C; angles , , are at A,B,C, respectively) by clicking the board at three positions. You do not have to be precise with your mouse: the system will find the nearest marked point automatically.
Then you can add more points by clicking any two existing points A and B. The system will create the associated point C.
In general the resulting set will be a dense subset of the plane. Can you find some such that the associated closure is not a dense subset of the plane? Can you find some such that is a subset of the given grid?
You can choose whether you let the system paint all triangles you create, or whether only their vertices (the points of ) are shown. The triangles sometimes create a nice artistic pattern on their own, especially when you turn the opacity down.
Click the "delete last" button to undo the latest addition.
Click the "reset" button to start from scratch.
The colors of the triangles are randomized.
You can choose between a rectangular background grid or a 60-degree (isometric) grid.
Click the "snap to grid" button to force the first three points (the seed triangle) to be positioned on the grid.
Examples (use the "closed set" drop-down menu):
Example 1 shows the seed that produces the -closed square lattice as a result.
Example 2 shows the seed whose -closed set is a subset of the square lattice. Can you figure out what looks like?
Example 3 shows the answer to the question of example 2: The -closure is a highly ordered subset of the square lattice. Can you actually reach all the points in from the given seed triangle?
Example 4 shows the seed (a regular triangle) that produces the -closed isometric lattice as a result.
In 1982 the author published a paper on (, )-closure and other closure types in the American Mathematical Monthly.
The prime results were:
a) The convex hull of a triangle-closed set is the plane.
b) If has sides , , opposite A, B, C such that , and , then every -closed set is a dense subset of the plane.
c) Let be an (, )-closed set. Then, in general, is a dense subset of the plane. The only exceptions occur for .
d) Let denote the square lattice that is the -closed hull of and , and let denote the isometric lattice that is the -closed hull of . Then each point C of or that does not lie on the axis defines an (, )-closure property that does not induce density.
Many of the various non-dense sets resulting from d) are far from trivial and interesting to study, and are the main reason for this Demonstration.
The other two topics of this paper were "mirror-closed" and "center-closed".
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