Study guide: tiling and crystallography

H. K. D. H. Bhadeshia

This guide explores the properties of the "hat" tile discovery, the mechanics of tessellation, and the scientific criteria required for a pattern to be classified as a crystal.

Part 1 & 2: interactive quiz

Instructions: review the questions below. Click "Show answer" to verify your understanding against the source text.

1. What is the physical description of the tile shape mentioned in the text?
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The tiles possess a clever shape that allows them to tessellate and fill a two-dimensional surface entirely. Each tile is identical in shape, resembling a shirt featuring a neck and two sleeves.
2. Who originally discovered the specific tile shape known as the "hat"?
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The shape was originally discovered by David Smith. Following its discovery, the "hat" tile was brought to the attention of the mathematics community in Cambridge.
3. According to the text, why can the tiles fill a two-dimensional surface completely?
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The tiles can fill space because they are designed with a shape that enables them to tessellate. To achieve this, individual tiles can be rotated relative to the axes of the wall to ensure the surface is fully covered.
4. How do individual tiles vary within the tessellation even if they share the same shape?
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While the shape of every tile is identical, they vary in their orientation relative to the axes of the surface. Additionally, tiles may be assigned different colours to create visual variation within the pattern.
5. What prevents two tiles of the same colour and orientation from being considered lattice points?
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Even if two tiles share the same orientation and colour, they are not lattice points if their surrounding environments are different. The text demonstrates this by noting that neighbouring tiles may have different colours or orientations, changing the local context.
6. Why does the tiling pattern described in the source not qualify as a crystal?
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The pattern does not qualify as a crystal because it lacks long-range periodicity. In crystallography, the absence of this periodic repetition prevents the arrangement from being classified as a crystal.
7. Where did Stoyan Smoukov implement this tiling pattern?
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Stoyan Smoukov utilised these specific tiles for a project at a sink in his house. He also provided the imagery used to illustrate the tiling concepts in the text.
8. What specific visual comparison is used to describe the geometric structure of the tiles?
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The text compares the shape of the tiles to a common article of clothing. Specifically, they are described as being shaped "rather like a shirt with a neck and two sleeves."
9. To whom did David Smith bring his discovery for further mathematical attention?
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After David Smith discovered the shape, it was brought to the attention of mathematicians in Cambridge. This move connected the amateur discovery with formal academic study in the field of mathematics.
10. Which academic institutions are associated with the provided source information?
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The source lists the School of Engineering and Materials Science at Queen Mary University of London and the Department of Materials Science & Metallurgy at the University of Cambridge.

Part 3: Essay topics

Use these prompts to explore the intersections of geometry, mathematics, and material science. Consider how the atomic structure of elements like iron or carbon compares to these macro-scale tiles.

Part 4: glossary

Term Definition
Crystallography The study of crystals and their structure; it involves the analysis of patterns and their periodicity.
Hat Tile The specific tile shape discovered by David Smith capable of filling a two-dimensional surface aperiodically.
Lattice Points Positions in a pattern with identical environments; identical orientation is insufficient if surroundings differ.
Long-range Periodicity A regular, repeating arrangement of units over a large area; defined mathematically as translation symmetry.
Tessellate To cover a 2D surface with shapes so there are no gaps or overlaps.
Two-dimensional Surface A flat area, such as a wall or sink splashback, used for tiling patterns.