Sierpinski Carpet Relatives Count

Details

squares removed	1	2	3	4	5	6	7
number of fractals	257	326	230	162	65	35	11

The subsquares are numbered as follows, and initially oriented in the up direction as illustrated by the arrows in the image on the right.

9 squares labels 1 to 9 from bottom to top 9 squares will arrows pointing up

We will restrict our count to Sierpinski Carpet relatives that are symmetric with respect to the vertical diagonal in the initial square only involve rotations (so no reflections). This means that squares 1, 5, and 9 can only be rotated by 0° or 180°. In addition, the squares in the pairs (2,3), (4,6), and (7,8) must either be both pointing up, both pointing down, or one pointing right and the other pointing left. Following is an example of these various symmetric possibilities.

9 squares with symmetric pointing arrows

So once a rotation is chosen for squares 2, 4, and 7, the rotations for squares 3, 6, and 8 are uniquely determined. We will also count two fractals as the same if one is a reflection of the other across the horizontal diagonal in the initial square.

Note: All fractals are shown using 4 iterations.

Remove 1
Square

If only 1 square is removed, it must be either square 1, 5, or 9.

Remove square 5

Removing square 5, the middle square, with no additional rotations produces the Sierpinski carpet.

111-1x1-111 design $111-1x1-111 fractal$

Because of the rotational symmetry of the fractal, the image can be rotated 0, 90, 180, or 270 degrees without changing the image. This means that each of the eight remaining squares in the construction can be rotated by any of these angles and the iterative construction will produce the same fractal. So removing square 5 produces only one possible fractal.

Remove square 9

Removing square 9, the top square, with no additional rotations produces the following fractal.

111-111-11x design $111-111-11x fractal$

Squares 1 and 5 can each be rotated in two ways (0° or 180°), while squares 2, 4, and 7 each have four choices. There are therefore $2^2\cdot4^3 = 256$ possible choices, and each produces a different fractal.

Remove square 1

Square 1 is symmetrically located relative to square 9. So each choice of rotations when square 9 is removed has a corresponding choice of rotations when square 1 is removed that will produce fractals that are just reflections of each other across the horizontal diagonal of the original square. For example, compare the two designs below, where the one on the left has square 9 removed and the one on the right has square 1 removed. Click on either fractal to see its reflected version and notice that both are the same when one is reflected.

111-111-11x design x11-111-111 design $x11-111-111 fractal$

Here is another example where three of the squares (4, 5, and 6) are rotated. Click on either fractal to see its reflected version and notice that both are the same when one is reflected.

111-234-11x design $111-234-11x fractal$ x11-432-111 design $x11-432-111 fractal$

This can be done for each of the 256 fractals obtained by removing square 9. Therefore deleting only square 1 does not produce any new fractals, just reflections of the ones from removing only square 9.

Total

The total number of fractals when removing just one square is therefore 1 + 256 + 0 = 257.

Remove 2
Squares

There are several cases for how two squares can be removed. As above, certain combinations can produce images that are just reflections of each other. Those will be counted together.

Remove squares 5,9 OR remove squares 1,5

Removing squares 5 and 9 with no additional rotations produces the following fractal on the left. The equivalent reflected version from removing squares 1 and 5 is shown on the right.

111-1x1-11x design $111-1x1-11x fractal$ x11-1x1-111 design $x11-1x1-111 fractal$

In either case, the remaining square 1 or 9 can be rotated in two ways (0° or 180°), while squares 2, 4, and 7 each have four choices. There are therefore $2\cdot4^3 = 128$ possible choices, and each produces a different fractal.

Remove squares 1,9

There are six cases to consider depending on how squares 2, 4, and 7 are rotated.

Case 1. Removing squares 1 and 9 with no additional rotations produces the following fractal.

x11-111-11x design $x11-111-11x fractal$

Any of the squares can be rotated by 0° or 180° without changing the fractal because of the 180° rotational symmetry of the image. For example, consider the following design where three of the squares are rotated by 180°.

x11-131-33x design $x11-131-33x fractal$

Therefore there is only one possible fractal if all squares are rotated by 0° or 180°.

Case 2. Now consider the fractal when only square 4 is rotated by 90° or 270°. Here square 4 is rotated by 90°, and thus square 6 is rotated by 270°.

x11-234-11x design $x11-234-11x fractal$

Again, because of the symmetry of the fractal, nothing will change if any of the squares with arrows pointing up are rotated by 180°. In addition, rotating the fractal by either 90° or 270° yields the same design, so it doesn't matter whether square 4 is rotated by 90° or 270°. Therefore there is only one possible fractal when square 4 is rotated.

Case 3. Next consider the fractal when squares 2 and 7 are rotated by 90° or 270°. Here they are both rotated by 90°, and thus squares 3 and 8 are rotated by 270°.

x24-111-24x design design $x24-111-24x fractal$

As before, because of the symmetry of the fractal, nothing will change if any of the squares with arrows pointing up are rotated by 180°. In addition, rotating the fractal by either 90° or 270° has the same result, so it doesn't matter whether squares 2 and 7 are rotated by 90° or 270°. Therefore there is only one possible fractal when squares 2 and 7 are rotated.

Case 4: There is one more symmetric case where squares 2, 4, and 7 are rotated by 90° or 270°. Here they are all rotated by 90°, and thus squares 3, 6, and 8 are rotated by 270°.

x24-214-24x design $x24-214-24x fractal$

In a similar way as above, because of the symmetry of the fractal this is the only possible image that can occur in this case regardless in how squares 2, 4, and 7 are rotated by either 90° or 270° and square 5 is rotated by 0° or 180°

Case 5. That takes care of all the symmetric cases. Now suppose square 7 is rotated by 90° or 270°. Here is the fractal if square 7 is rotated by 90°, and so square 8 is rotated by 270°.

x11-111-24x design $x11-111-24x fractal$

Because of the lack of symmetry, rotating any of the squares with arrows pointing up by 180° will produce a different fractal, and rotating square 7 by 270° will also produce a different fractal. There are two rotation choices for each of the squares 2, 4, 5, and 7, so the total number of possible fractals in this case is $2^4 = 16$. Each of these 16 fractals has a reflected version obtained by rotating square 2 by 90° or 270° instead of square 7. Here is the reflected companion of the fractal shown above. Click on the fractal to reflect it across the horizontal line.

x24-111-11x design $x24-111-11x fractal$

Case 6. The final case is when squares 4 and 7 are rotated by 90° or 270°. Here is the fractal when both are rotated by 90°.

x11-214-24x design $x11-214-24x fractal$

Because of the lack of symmetry, there are two possible rotation choices for squares 4 and 7, and two possible rotation choices for squares 2 and 5, for a total of $2^4 = 16$ different fractals. Each of these 16 fractals has a reflected version obtained by rotating squares 2 and 4 by 90° or 270° instead of squares 4 and 7.

This exhausts all possible cases when squares 1 and 9 are removed. Adding up the different number of fractals gives 1+1+1+1+16+16 for a total of 36.

Remove squares 4,6

There are three cases depending on how squares 2 and 7 are rotated.

Case 1. Removing squares 4 and 6 with no additional rotations produces the following fractal.

111-x1x-111 design $111-x1x-111 fractal$

Because of the symmetry of the fractal, any of the squares can be rotated by 0° or 180° without changing the image. Therefore there is only one possible fractal if all squares are rotated by 0° or 180°.

Case 2. Consider when square 7 is rotated by 90° or 270°. Here square 7 is rotated by 90° (and thus square 8 is rotated by 180°).

111-x1x-241 design $111-x1x-241 fractal$

Because of the lack of symmetry, rotating any of the squares with arrows pointing up by 180° will produce a different fractal, and rotating square 7 by 270° will also produce a different fractal. There are two rotation choices for square 7 and also two rotation choices for each of the squares 1, 2, 5, and 9, so the total number of possible fractals in this case is $2^5=32$. Each of these 32 fractals has a reflected version obtained by rotating square 2 by 90° or 270° instead of square 7.

Case 3. For the final case, consider when both square 2 and 7 are rotated by 90° or 270°. Here square 7 is rotated by 90° (and thus square 8 is rotated by 270°), and square 2 is rotated by 270° (and thus square 3 is rotated by 90°).

142-x1x-241 design $142-x1x-241 fractal$

As in other cases above, because of the symmetry of the fractal this is the only possible image that can occur in this case regardless in how squares 2 and 7 are rotated by either 90° or 270° and squares 1, 5, and 9 are rotated by 0° or 180°.

This exhausts all possible cases when squares 1 and 9 are removed. Adding up the different number of fractals gives 1+32+1 for a total of 34.

Remove squares 7,8 OR remove squares 2,3

Removing squares 7 and 8 with no additional rotations produces the following fractal on the left. The equivalent reflected version from removing squares 2 and 3 is shown on the right.

111-111-xx1 design $111-111-xx1 fractal$ 1xx-111-111 design $1xx-111-111 fractal$

In either case, the remaining squares 1, 5, and 9 can be rotated in two ways (0° or 180°), while squares 2 and 4, or 4 and 7 for the reflected version, each have four choices. There are therefore $2^3\cdot4^2 = 128$ possible choices, and each produces a different fractal.

Total

The total number of fractals when removing two squares is therefore 128+36+128+34=326.

Remove 3
Squares

There are several cases for how three squares can be removed. As above, certain combinations can produce images that are just reflections of each other. Those will be counted together.

Remove squares 1, 5, and 9

This is very similar to the situation for removing two squares where squares 1 and 9 are removed. So the six cases illustrated above for that situation still apply, but square 5 will no longer be involved. The symmetric cases still give only one fractal, whereas the two non-symmetric designs will now only produce 8 fractals each instead of 16. Therefore there will be a total of 1+1+1+1+8+8=20 difference fractals. Here are two examples.

x11-1x1-11x design $x11-1x1-11x fractal$ x11-1x1-24x design

Remove squares 4, 5, and 6

This is very similar to the situation for removing two squares where squares 4 and 6 are removed. So the three cases illustrated above for that situation still apply, but square 5 will no longer be involved. The symmetric cases still give only one fractal, whereas the one nonsymmetric design will now only produce 16 fractals each instead of 32. Therefore there will be a total of 1+16+1=18 difference fractals.

Remove squares 7,8,9 OR remove squares 1,2,3

Removing squares 7, 8, and 9 with no additional rotations produces the following fractal.

111-111-xxx design $111-111-xxx fractal$

The remaining squares 1 and 5 can be rotated in two ways (0° or 180°), while squares 2 and 4 each have four choices. There are therefore $2^2 \times 4^2 = 64$ possible choices, and each produces a different fractal. Each of these has a corresponding reflected version obtained by removing squares 1, 2, and 3.

Remove squares 4,6,9 OR remove squares 1,4,6

Removing squares 4, 6, and 9 with no additional rotations produces the following fractal.

111-x1x-11x design $111-x1x-11x fractal$

The remaining squares 1 and 5 can be rotated in two ways (0° or 180°), while squares 2 and 7 each have four choices. There are therefore $2^2 \times 4^2 = 64$ possible choices, and each produces a different fractal. Each of these has a corresponding reflected version obtained by removing squares 1, 4, and 6.

Remove squares 2,3,9 OR remove squares 1,7,8

Removing squares 2, 3, and 9 with no additional rotations produces the following fractal.

1xx-111-11x design $1xx-111-11x fractal$

The remaining squares 1 and 5 can be rotated in two ways (0° or 180°), while squares 4 and 7 each have four choices. There are therefore $2^2 \times 4^2 = 64$ possible choices, and each produces a different fractal. Each of these has a corresponding reflected version obtained by removing squares 1, 7, and 8.

Total

The total number of fractals when removing two squares is therefore 20+18+64+64+64=230.

Remove 4
Squares

There are several cases for how four squares can be removed. As above, certain combinations can produce images that are just reflections of each other. Those will be counted together.

Remove squares 2,3,7,8

Removing squares 2, 3, 7, and 8 with no additional rotations produces the box fractal rotated by 45°.

1xx-111-xx1 design $1xx-111-xx1 fractal$

Because of the rotational symmetry of the fractal, the image can be rotated 0, 90, 180, or 270 degrees without changing the image. This means that each of the five remaining squares in the construction can be rotated by any of these angles and the iterative construction will produce the same fractal. So removing these four squares produces only one possible fractal.

Remove squares 1,4,6,9

Removing squares 1, 4, 6, and 9 with no additional rotations produces the following image.

x11-x1x-11x design $x11-x1x-11x fractal$

Remove squares 4,5,6,9 OR remove squares 1,4,5,6

Removing squares 4, 5, 6, and 9 with no additional rotations produces the following fractal.

111-xxx-11x design $111-xxx-11x fractal$

The remaining square 1 can be rotated in two ways (0° or 180°), while squares 4 and 7 each have four choices. There are therefore $2 \times 4^2 = 32$ possible choices, and each produces a different fractal. Each of these has a corresponding reflected version obtained by removing squares 1, 4, 5, and 6.

Remove squares 5,7,8,9 OR remove squares 1,2,3,5

Removing squares 5, 7, 8, and 9 with no additional rotations produces the following fractal.

111-1x1-xxx design $111-1x1-xxx fractal$

The remaining square 1 can be rotated in two ways (0° or 180°), while squares 2 and 4 each have four choices. There are therefore $2 \times 4^2 = 32$ possible choices, and each produces a different fractal. Each of these has a corresponding reflected version obtained by removing squares 1, 2, 3, and 5.

Remove squares 1,5,7,8 OR remove squares 2,3,5,9

Removing squares 1, 5, 7, and 8 with no additional rotations produces the following fractal.

x11-1x1-xx1 design $x11-1x1-xx1 fractal$

The remaining square 9 can be rotated in two ways (0° or 180°), while squares 2 and 4 each have four choices. There are therefore $2 \times 4^2 = 32$ possible choices, and each produces a different fractal. Each of these has a corresponding reflected version obtained by removing squares 2, 3, 5, and 9.

Remove squares 1,7,8,9 OR remove squares 1,2,3,9

Removing squares 1, 7, 8, and 9 with no additional rotations produces the following fractal.

x11-111-xxx design $x11-111-xxx fractal$

The remaining square 5 can be rotated in two ways (0° or 180°), while squares 2 and 4 each have four choices. There are therefore $2 \times 4^2 = 32$ possible choices, and each produces a different fractal. Each of these has a corresponding reflected version obtained by removing squares 1, 2, 3, and 9.

Remove squares 4,6,7,8 OR remove squares 2,3,4,6

Removing squares 4, 6, 7, and 8 with no additional rotations produces the following fractal.

111-x1x-x1x design $111-x1x-x1x fractal$

The remaining squares 1, 5, and 9 can be rotated in two ways (0° or 180°), while square 2 has four choices. There are therefore $4 \times 2^3 = 32$ possible choices, and each produces a different fractal. Each of these has a corresponding reflected version obtained by removing squares 2, 3, 4, and 6.

Total

The total number of fractals when removing four squares is therefore 1+1+32+32+32+32+32=162.

Remove 5
Squares

There are several cases for how five squares can be removed. As above, certain combinations can produce images that are just reflections of each other. Those will be counted together.

Remove squares 2,3,5,7,8

Removing squares 2, 3, 5, 7, and 8 with no additional rotations produces the following fractal.

1xx-1x1-xx1 design $1xx-1x1-xx1 fractal$

Remove squares 4,6,7,8,9 OR remove squares 1,2,3,4,6

Removing squares 4, 6, 7, 8, and 9 with no additional rotations produces the following fractal.

111-x1x-xxx design $111-x1x-xxx fractal$

The remaining squares 1 and 5 can be rotated in two ways and square 2 can be rotated in four ways. There are therefore $4 \times 2^2 = 16$ possible choices, and each produces a different fractal. Each of these has a corresponding reflected version obtained by removing squares 1, 2, 3, 4, and 6.

Remove squares 1,5,7,8,9 OR remove squares 1,2,3,5,9

Removing squares 1, 2, 3, 5, and 9 with no additional rotations produces the following fractal.

xxx-1x1-11x design $xxx-1x1-11x fractal$

The remaining squares 2 and 4 can be rotated in four ways. There are therefore $4^2 = 16$ possible choices, and each produces a different fractal. Each of these has a corresponding reflected version obtained by removing squares 1, 5, 7, 8, and 9.

Remove squares 2,3,7,8,9 OR remove squares 1,2,3,7,8

Removing squares 2, 3, 7, 8, and 9 with no additional rotations produces the following fractal.

1xx-111-xxx design $1xx-111-xxx fractal$

The remaining squares 1 and 5 can be rotated in two ways (0° or 180°), while square 4 has four choices. There are therefore $4^2 = 16$ possible choices, and each produces a different fractal. Each of these has a corresponding reflected version obtained by removing squares 1, 2, 3, 7, and 8.

Remove squares 4,5,6,7,8 OR remove squares 2,3,4,5,6

Removing squares 4, 5, 6, 7, and 8 with no additional rotations produces the following fractal.

111-xxx-xx1 design $111-xxx-xx1 fractal$

The remaining squares 1 and 9 can be rotated in two ways (0° or 180°), while square 2 has four choices. There are therefore $4^2 = 16$ possible choices, and each produces a different fractal. Each of these has a corresponding reflected version obtained by removing squares 2, 3, 4, 5, and 6.

Total

The total number of fractals when removing five squares is therefore 1+16+16+16+16=65.

Remove 6
Squares

There are several cases for how six squares can be removed. As above, certain combinations can produce images that are just reflections of each other. Those will be counted together.

Remove all squares except squares 1,5,9

Removing all squares except for squares 1, 5, and 9 without rotating any of the remaining squares produces the following image (just a vertical line segment, so technically not really a fractal).

1xx-x1x-xx1 design $1xx-x1x-xx1 fractal$

Any of the remaining three squares can be rotated by 180° without changing the final image, so there is only one possibility in this case.

Remove all squares except squares 4,5,6

Removing all squares except for squares 4, 5, and 6 without rotating any of the remaining squares produces the following image (just a horizontal line segment, so technically not really a fractal).

xxx-111-xxx design $xxx-111-xxx fractal$

The image will not change if any of the squares are rotated by 180° If square 4 is rotated by 270° (and thus square 6 by 90°), the result is shown below.

xxx-412-xxx design $xxx-412-xxx fractal$

The same fractal would result whether square 5 is pointing up or down, and whether square 4 is pointing left or right. Therefore these two images are the only possibilities when all squares except 4, 5, and 6 are removed.

Remove all squares except squares 1,2,3 OR except 7,8,9

Removing all squares except for squares 1, 2, and 3 without rotating any of the remaining squares produces the following fractal.

111-xxx-xxx design $111-xxx-xxx fractal$

The remaining square 1 can be rotated in two ways (0° or 180°), while square 2 has four choices. There are therefore 4×2 = 8 possible choices, and each produces a different fractal. Each of these has a corresponding reflected version obtained by removing all squares except 7, 8, and 9.

The following three cases are similar to the one above, and each has 8 possible choices.

Remove all squares except squares 1,4,6 OR except 4,6,9

1xx-1x1-xxx design $1xx-1x1-xxx fractal$

Remove all squares except squares 1,7,8 OR except 2,3,9

1xx-xxx-11x design $1xx-xxx-11x fractal$

Remove all squares except squares 2,3,5 OR except 5,7,8

x11-x1x-xxx design $x11-x1x-xxx fractal$

Total

The total number of fractals when removing five squares is therefore 1+2+8+8+8+8=35.

Remove 7
Squares

Following are the four basic designs from removing 7 squares and not rotating the remaining two.

1xx-xxx-xx1 design xxx-1x1-xxx design 1xx-x1x-xxx design x11-xxx-xxx design

The first produces a vertical Cantor set. The second produces a horizontal Cantor set. The other two produce Cantor like sets. Including rotations, there are a total of 1+2+4+4=11 different possible fractals from the four designs (and their reflective companions).

See a Gallery of some of the Sierpinski Carpet relatives.
See some video demonstrations of the IFS construction of some of the Sierpinski Carpet relatives.

Classic Iterated Function Systems