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*.

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".