Is it possible to tile the plane with circles

The image should be at least 4 tiles wide and 4 tiles tall. The image above is 4 tiles by 4 tiles. The tessellation may be translated by any amount. For example, the top left corner of the image does not need to be the vertex where tiles meet. The tessellation should not be rotated, however.

You may use external graphics libraries that have commands for drawing circles and outputting images and the like. The curves really should approximate circles, as can done with the midpoint circle algorithm , which most graphics libraries will do for you. I am using as many default parameters for camera and lightsource as possible, thats why it is a little dark.

Lets ungolf it first. It is obvious what is happening once we increase the offset of the cylinder axis and change the perspective. I noticed that the boundaries between the cells look very sinusoidal, so i went for an analytical solution with a very simple core equation. While it looks similar, the circles are way too square. You can see the result at the Context Free Gallery. Thanks to the infinite alephalpha for a clever array-based method:.

That boolean is used as the conditional to a pattern that replaces the whole 'white' tile with the transformed, 'black' tile. From there, ArrayFlatten joins together the tiles and Image displays them. Note the use of the shorter Thread to replace Transpose. We can still save 4 bytes by using the transpose symbol instead. In UTF-8 that brings the total to 90 bytes in 88 characters.

If the plane is a chessboard, these are the 'white' squares. For the 'black' squares, we need to invert the colors and rotate by 90 degrees. If the dimensions of the image are even like the minimum size, 4 , then a tile on the right edge will be the same as the next one on the left edge. However, adding one tile to get an odd size 5 then concatenating the rows produces a regular alternating pattern.

This suggests that we can get the full image by wrapping a single row of alternating tiles with Partition. Then we Partition the list into a 5 by 5 matrix of tiles Partition discards the trailing 26 th tile :. Finally ArrayFlatten turns the matrix of tile matrices into a flat matrix, and Image displays the result. Applet: A small application program that can be called up for use while working in another application. How did you hear about us? Consent to receive advertising text messages and calls is not required to purchase other goods or services.

There are three different types of tessellations source : Regular tessellations are composed of identically sized and shaped regular polygons. Semi-regular tessellations are made from multiple regular polygons. Only eight combinations of regular polygons create semi-regular tessellations.

Meanwhile, irregular tessellations consist of figures that aren't composed of regular polygons that interlock without gaps or overlaps. As you can probably guess, there are an infinite number of figures that form irregular tessellations! Hours of Instruction. The cube is a tiling of a 2-sphere by squares, meeting 3 at each vertex It isn't the same tiling, but I allowed that with the trivial one I mentioned.

You also get triangles 3,4,5 at a vertex. And your tiling with pentagons also doesn't correspond to any tiling of the plane, so I presumed you were already working under different rules.

And of course, it's still the case that you can't tile the sphere with hexagons regardless unless you use the trivial tiling, because they can't meet fewer than 3 at a vertex, and that tiling is planar rather than spherical. Sign up or log in Sign up using Google. Sign up using Facebook. Sign up using Email and Password. Post as a guest Name. Email Required, but never shown. Upcoming Events.

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