Sphinx tiling
A pentagonal rep-tile tiling the plane non-periodically.
The sphinx tiling covers the plane with a single pentagonal shape known as the sphinx, which is built from six equilateral triangles. Its name comes from its likeness to the Great Sphinx of Giza. Because a sphinx can be cut into any square number of smaller sphinxes (some of which are mirror images), and repeating this process yields a non-periodic tiling, the shape is classified as a rep-tile. It is one of only a few pentagonal rep-tiles, and the only one where all the smaller copies are the same size.
Beyond recursive tilings, a sphinx-shaped outline—called a frame—can be tiled in non-recursive ways for any order. The order is defined by the number of triangles at the tail end on a triangular lattice. An order‑2 frame has exactly two distinct tilings using four sphinxes, while an order‑3 frame can be tiled with nine sphinxes in four different ways. The number of possible tilings grows rapidly with the square of the order, though an exact growth rate is not widely established.
- field
- Geometry
- known_for
- Pentagonal rep-tile tessellation of the plane
- type
- Tessellation
- shape
- Sphinx (pentagonal hexiamond)
Lore & Background
The sphinx tiling is based on the sphinx shape, a pentagonal hexiamond made from six equilateral triangles. This shape can be dissected into any square number of copies of itself, some of which are mirror images. Repeating this process yields a non-periodic tiling of the plane, making the sphinx a rep-tile. It is one of few known pentagonal rep-tiles and the only one whose sub-copies are equal in size.
Reader's Guide
The sphinx tiling is significant in geometry as a rare example of a pentagonal rep-tile. Its ability to be dissected into square numbers of smaller copies, including mirror images, allows for recursive self-replication that produces a non-periodic plane tiling. This property distinguishes it from many other tilings. Additionally, an outer boundary in the shape of a sphinx (a 'frame') can be tiled non-recursively for all orders, where the order is defined by the number of triangles at the tail end. For example, an order-2 frame can be tiled by four sphinxes in exactly one way, while an order-3 frame can be tiled by nine sphinxes in four ways. The number of such tilings grows exponentially with the order, approximately as e^{c n^2} with c ≈ 0.425. This exponential growth highlights the combinatorial complexity of sphinx tilings, making them a subject of study in tessellation theory.
Did You Know?
- The sphinx shape is a pentagonal hexiamond formed by gluing six equilateral triangles together.
- A sphinx can be dissected into any square number of copies of itself, some of which are mirror images.
- The sphinx is the only known pentagonal rep-tile whose sub-copies are equal in size.
- An order-2 sphinx frame can be tiled by four sphinxes in exactly two distinct ways.
Frequently Asked Questions
What is the Sphinx tiling?
It is a non-periodic tessellation of the entire plane built from a single pentagonal shape called the sphinx. The pattern fills every point without gaps or overlaps yet never repeats in a simple grid.
What is the sphinx shape made of?
The sphinx is a pentagonal hexiamond, assembled from six equilateral triangles joined edge-to-edge. Its overall silhouette is what inspired the name, echoing the profile of the Great Sphinx of Giza.
Why is the sphinx classified as a rep-tile?
A single sphinx can be dissected into any perfect-square number of smaller sphinxes, some of which are mirror images. Iterating that subdivision indefinitely produces a non-periodic tiling, which is the defining property of a rep-tile.
What makes the sphinx special among pentagonal rep-tiles?
It is one of only a handful of known pentagonal shapes that qualify as rep-tiles. What sets it apart is that every smaller copy produced in the subdivision is the same size, a property no other pentagonal rep-tile shares.
Can the sphinx tiling be done without recursion?
Yes—beyond the self-similar recursive construction, a sphinx-shaped outline (often called a frame) can also be filled with smaller sphinxes in non-recursive arrangements. This shows the shape is versatile well beyond its most famous infinite pattern.
More in Sphinxes 1-24
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