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Penrose Tiling Quilt Patterns

Updated on January 20, 2017
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TR Smith is a product designer and former teacher who uses math in her work every day.

A Penrose tiling—named for its discoverer Roger Penrose—is a non-periodic quasicrystal tiling pattern based on a regular pentagon. A non-periodic or aperiodic tiling lacks translational symmetry; a quasicrystal tiling is ordered. Since a Penrose tiling doesn't repeat at regular intervals, there are an infinite number of unique quilt designs you can create with Penrose tiles. One of the most basic forms of this tiling uses two rhombus shapes, one wider and one narrower. The wider rhombus has angles of 72° and 108°, while the narrower rhombus has angles of 36° and 144°. These are shown in the diagram below.

You can arrange the basic tiles to create 5-pointed star and 5-petaled flower motifs, or any other interesting shapes. Here are some examples of quilt patterns using Penrose tiles, along with some basic quilting math for sewing beautiful Penrose quilts.

Two rhombuses of equal side lengths with different angles.
Two rhombuses of equal side lengths with different angles.

Piecing the Rhombuses Together

Let's call the fatter rhombus "A" and the skinnier rhombus "B." In the figure above, A is the yellow shape and B is the blue one. To make a Penrose tiling, both A and B have to have the same side length. With that condition met, there are many ways to make a non-periodic patchwork tessellation of rhombuses.

Notice that all the angles in these rhombuses are divisible by 36. That is, 72 = 2*36, 108 = 3*36, 36 = 1*36, and 144 = 4*36. The total angle of the plane is 360 degrees, which is 10*36. When piecing together the rhombuses to form a solid pattern without gaps, the total angle of the corners that meet at a point must equal 360 degrees.

If you imagine rhombus A as having corners labeled {2, 3, 2, 3} (since 2*36 = 72 and 3*36 = 108), and rhombus B as having corners labeled {1, 4, 1, 4} (since 1*36 = 36 and 4*36 = 144), then the sum of the rhombus corners that meet at a point must equal 10. This is illustrated in the diagram below with 6 examples.

For example, the arrangement in the middle of the top row works because 2 + 2 + 2 + 4 = 10. These are but a few examples of valid arrangements of rhombuses in a Penrose tiling. Once you know this basic principle, you can make your Penrose quilt designs.

Penrose Pattern Gallery

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Penrose Quilting Math

If the side lengths of rhombuses A and B are L, then the areas of the rhombuses are

Area of A = sin(72°) * L^2 ≈ 0.95106 * L^2
Area of B = sin(36°) * L^2 ≈ 0.58779 * L^2

If the number of A rhombuses used in the quilt is M, and the number of B rhombuses used in the quilt is N, then the area of the quilt, not including the border, is given by the approximation formula

Quilted Area ≈ (0.95106M + 0.58779N) * L^2

For example, suppose you construct a Penrose quilt with rhombuses whose side length is 3.5 inches. You use 103 of type A and 62 of type B, then the approximate area of the quilt is

(0.95106*103 + 0.58779*62) * 3.5^2
= 1646.42646 square inches
≈ 11.43 square feet.

How Many Rhombuses to Use?

Suppose you want to sew a quilt with an area of X square inches using rhombuses that are L inches along the sides. How many rhombuses of type A and B do you need? One thing to know about rhombic Penrose tilings is that the ratio of fat rhombuses to thin rhombuses approaches the golden mean (sqrt(5) + 1)/2 ≈ 1.618 as the pattern spreads out. (See this article for more mathematical detail) This means on average there are about 1.618 times more of the type A rhombuses than type B, so M = 1.618N.

Using this, you can figure the approximate number of each type of rhombus for a Penrose quilt. The formulas are

Number of type A rhombuses ≈ 0.760845 * X / L^2
Number of type B rhombuses ≈ 0.470228 * X / L^2

For example, suppose you want to make a Penrose quilt that is about 1650 square inches in area using rhombuses that are 3.5 inches along the sides. Then the approximate number of A and B rhombuses you need are computed by

# type A ≈ 0.760845 * 1650 / 3.5^2
≈ 102.5

# type B ≈ 0.470228 * 1650 / 3.5^2
≈ 63.3

This includes fractional pieces of rhombuses if you cut it to a rectangle.

See also Best Calculation Tools for Quilters.

Penrose tiling patchwork quilt
Penrose tiling patchwork quilt

Other Forms of Penrose Tilings

The aperiodic rhombus tiling is but one variation of a Penrose tiling. Another common form is the kite and dart tiling, which as its name suggests, uses a kite shape and a dart shape to create a non-periodic tessellation. Some other types Penrose tilings are shown in the gallery below.

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Non-Penrose Arrangements of Rhombuses

The A and B rhombuses can be arranged to form other tessellations that are not classified as Penrose tilings. In the figure below, the arrangement on the left is non-Penrose because it is periodic. The arrangement on the right is aperiodic, but it is not quasicrystal, thus it is not considered a Penrose tiling either. Both can still be used to make interesting quilt patterns.

These rhombic tilings are not Penrose tilings.
These rhombic tilings are not Penrose tilings.

Here are two more arrangements of rhombus tiles that are periodic, thus not Penrose tilings, and suitable for quilting projects.

Periodic rhombus tiling with Penrose tiles
Periodic rhombus tiling with Penrose tiles
A periodic rhombus tiling that is not a Penrose tiling.
A periodic rhombus tiling that is not a Penrose tiling.

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    • Millionaire Tips profile image

      Shasta Matova 3 years ago from USA

      Wow - what beautiful quilts! These are not simple quilts with all those y seams, but definitely worth the extra effort!

    • calculus-geometry profile image
      Author

      TR Smith 3 years ago

      It's definitely a time-consuming project! I think the hard part would be cutting the shapes precisely and piecing everything together very very very precisely. It's something I've had on the back burner for a while.

    • tirelesstraveler profile image

      Judy Specht 3 years ago from California

      I am going to share this. Wow.

    • calculus-geometry profile image
      Author

      TR Smith 3 years ago

      Thanks, it's one of my favorite quilting patterns, but not something you see too often.

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      grace 2 years ago

      just love the unusual

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