In the house I was staying in while in Bellevue/WA, there was a floor pattern that looks like the image below. It's easy to figure out how to create such a pattern using a combination of small and big tiles. I tried to create that pattern in a graphics program just for fun.
I used Inkscape to create the image below but it turned out to be a tedious job. It's really boring to clone figures and align them in that shape. The right way to do it is to use patterns. It's not trivial, however, to figure out the pattern that repeats. Note that a repeating pattern should be a rectangular area. Two squares of different sizes do not form a valid pattern to use.
The easy way is to find a large rectangular area that repeats. This repeating pattern can be easily found and highlighted as shown below. However, it doesn't really seem nice to have such a complicated pattern. Moreover, according to the sizes of the two squares, the repeating pattern could become arbitrarily and extremely large. An alternative way to make it is described in this post.
The trick to find the repeating pattern is to tilt your head a little bit. The pattern really repeats along a line with a small angle. If we align small squares on a horizontal line, the pattern would contain one small square. All we need to do is to tilt the whole view with the correct angle. In the shown pattern, square sizes are 10 and 40. Therefore, the right angle is atan(10/40) ≅ 14°. Rotating the view brings the following image.
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| Rotate all drawn figures |
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| How it looks after rotation |
Once the view is tilted as shown above, any square of the correct size would form a valid repeating pattern. The correct size is a square enclosing the complete repeating pattern. Simply, the borders of this repeated pattern would span from any point, to its counterpart in the adjacent tile. Let's take the corners of the small black square as our pivot point. The repeating pattern will be similar to the one below.
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| A sample part of the image that can be used as a pattern. Note: I changed the black squares into holes just for presentation and clearance. |
You can easily use triangle geometry to calculate the size of the repeating tile (it's a square by the way). From the figure below, the length of the edge of this square is sqrt(10
2+40
2).
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| How to calculate the size of the pattern square |
Once we find the size of the repeating pattern, cropping any square of this size from the figure we created forms a repeating pattern. Note that, by definition, the pattern repeats and logically it has no begin or end. We create a (pattern) square of the calculated size above. Then, we union all the repeating (rotated) figure we created earlier. Finally we place the (pattern) square any place on the figure and find the intersection of both. All these steps are shown in figures below.
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| Create a square to be used as the pattern |
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| Union the figure we created earlier. Note that we need this figure to contain only one black square. |
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| Intersection the pattern square with the figure we drew |
This leaves us with the pattern shown in the figure below. Using this a tiled pattern would produce the same (rotated) pattern we drew earlier. You can then rotate the object that contains the pattern to produce the original figure we wanted to draw.
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| The resulting pattern |
The final view will look similar to the one below.
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| Final view drawn using the pattern |
Note that the final view has some glitches. The reason is that I used Inkscape which limits all numbers I use (for rotation degrees, lengths ... etc.) to three digits after the decimal point. This means that the accuracy is limited to this precision and I couldn't create a very accurate figure. I guess that another program may overcome this problem but the main idea is still valid.
I used Inkscape to create the image below but it turned out to be a tedious job. It's really boring to clone figures and align them in that shape. The right way to do it is to use patterns. It's not trivial, however, to figure out the pattern that repeats. Note that a repeating pattern should be a rectangular area. Two squares of different sizes do not form a valid pattern to use.
The easy way is to find a large rectangular area that repeats. This repeating pattern can be easily found and highlighted as shown below. However, it doesn't really seem nice to have such a complicated pattern. Moreover, according to the sizes of the two squares, the repeating pattern could become arbitrarily and extremely large. An alternative way to make it is described in this post.
The trick to find the repeating pattern is to tilt your head a little bit. The pattern really repeats along a line with a small angle. If we align small squares on a horizontal line, the pattern would contain one small square. All we need to do is to tilt the whole view with the correct angle. In the shown pattern, square sizes are 10 and 40. Therefore, the right angle is atan(10/40) ≅ 14°. Rotating the view brings the following image.
