Tessellation: Roman Roads, Escher, and the Modern Sky Map

Do you know the Dutch artist M. C. Escher? He built a career on shapes that interlock perfectly, no gaps, no leftover space — fish becoming birds, lizards becoming diamonds, with no visible seam where one figure ends and the next begins.

You don't need a gallery ticket to see this idea in action. Step outside — the sidewalk under your feet is probably doing the same thing, rectangles or hexagons repeating with no gap. Plainer than Escher's lizards, same principle.

That principle has a name: tessellation, and people were using it long before Escher or the concrete plant that paved your sidewalk. One of the earliest large-scale examples: a road out of Rome, begun roughly 2,000 years ago.

The rule in one picture: gaps and overlaps don't count — only edge-to-edge, no-gap coverage is a true tessellation.

The Appian Way was begun in 312 BCE and eventually ran hundreds of miles from Rome to Brundisium. It was built about 20 feet wide and slightly convex, so rainwater drained toward the edges. On top sat polygonal blocks of volcanic lava, laid over a mortared stone foundation. Roman poets, among them Horace, called it the "queen of long-distance roads," and Encyclopaedia Britannica credits that lava surface with the road's extraordinary durability.

Old photographs of the surviving stones show why. The blocks weren't a tidy hexagonal grid — they were irregular shapes, smoothly fitted so each one locked against its neighbors. It's the same impulse Escher would later turn into art: divide a surface so nothing is wasted and nothing shifts loose.

Modern interlocking pavers on a city sidewalk — the same logic Roman engineers relied on in antiquity.

Tessellation connects M. C. Escher's impossible art, the stone roads of ancient Rome, and the way astronomers now map the sky. This article traces how one mathematical idea — covering a surface with repeating shapes and no gaps — has quietly shaped engineering, art, and science for two millennia.

The Romans Solved the Tessellation Problem First

Roman engineers built for permanence, and their pavement choices show it. Historians call these floors "tessellated pavements" — small stone cubes fitted closely into geometric or figurative designs. They first appeared in the Hellenistic period, then spread across the empire by the first century CE. Builders prized them for two things: durability, and precise visual detail.

One example makes the detail argument impossible to dismiss: the Battle of Issus mosaic from Pompeii's House of the Faun — now in Naples — reproduces Alexander the Great's clash with Darius III of Persia in tessellated stone at floor level, piece by tiny piece.

That's where the standard account usually stops — and, to my eye, where it misses the point. The geometry wasn't chosen for beauty first. Roman pavers sized their pieces to fit tightly against one another, because that tiled a surface completely and, set in mortar, held up against wear — the same reasoning, scaled up, behind the Appian Way's fitted lava surface.

Today's concrete pavers carry that logic forward with new materials. Rectangles, squares, and hexagons remain the default because tightly fitted units transfer load sideways into their neighbors, and one damaged piece can be swapped without disturbing the rest. The problem hasn't changed since.

From Roman road layers to modern urban paving: the structural logic behind tessellation hasn't needed an update in two millennia.

The Artist Who Took Tessellation Off the Ground

Maurits Cornelis Escher was born in the Netherlands in 1898 and became one of the world's most recognized graphic artists through lithographs, woodcuts, and wood engravings exploring impossible architecture and interlocking shapes. His first tessellating artwork dates to the 1920s, later expanded into a series called "Regular Division of the Plane": animals and figures fitted together like puzzle pieces, no gaps, no overlaps.

He was not a trained mathematician. That's the part that strikes me most.

Mathematician H. S. M. Coxeter, known for his work in hyperbolic geometry, recognized real geometric intuition in Escher's prints. He shared diagrams of hyperbolic tilings — grids where shapes shrink progressively toward a circle's edge without ever reaching it. Those diagrams fed directly into Escher's circle-limit series, where figures diminish toward the boundary, implying infinity within a finite frame.

Coxeter had the formal training Escher lacked — and found that Escher had gotten there anyway, through observation alone.

In the circle-limit prints, tessellation stops being a floor pattern. It becomes a visual argument about the nature of space itself — a geometry where parallel lines diverge, and the boundary represents an infinity you can see but never reach. Escher put advanced mathematics on a wall, no equations required.

A hyperbolic tiling built on Escher's method, carrying the same shrink-toward-the-edge geometry into later mathematical art.

Not every insight finds a Coxeter. Read about a discovery that went unrecognized for decades: Ida Noddack: the scientist who discovered element 43 — and was ignored

Tessellation Everywhere: From Algorithms to the Night Sky

It shows up in places I'd never think to look for a floor pattern.

When engineers simulate airflow over a wing or fluid through a pipe, they divide the space into a computational mesh — small cells that together tessellate the domain. The same technique let DreamWorks animators compute the water and smoke effects in "The Prince of Egypt" using real fluid-dynamics equations. Triangular and hexagonal cells get chosen for the same reason a Roman paver chose a cube: they tile without gaps, though a stretched cell can still throw off the whole simulation.

Architecture works the same way, at building scale instead of cell scale. Islamic geometric design is a clear case: its star-and-polygon patterns are regular and semiregular tessellations, repeating polygons tiling a surface with no gaps. Contemporary architects use the same vocabulary for glass roofs and façades, translating hexagonal grids — sometimes borrowed from honeycomb cells — into structural panels.

Urban paving patterns — from rectangular grids to hexagonal tiles — illustrate how tessellation principles shape contemporary public space design.

Tessellation Beyond Earth: Mapping the Sky

Zoom out far enough, and astronomers face the same problem mapping the sky itself: cover the celestial sphere without gaps or overlaps — tessellation wrapped around a globe instead of laid flat. NASA's Nancy Grace Roman Space Telescope solves it with a scheme called HEALPix — short for Hierarchical Equal Area isoLatitude Pixelization, a mouthful, but the idea is simple. The sphere divides into twelve diamond-shaped base tiles. Each one splits into four smaller tiles, over and over, so every pixel ends up covering exactly the same area, no matter where it falls. Even the earliest known star catalogs, compiled by Babylonian astronomers over three thousand years ago, were already an attempt to organize the whole sky into one systematic listing. The instruments changed. The geometry didn't.

What strikes me as strange is that the same geometric decision keeps resurfacing in contexts with no obvious connection. A Roman mosaic floor, a fluid-dynamics mesh, and a space telescope's sky archive share no lineage, yet all answer the identical question: how do you cover a surface so nothing goes to waste?

Once you notice the pattern, the ordinary is hard to walk past without seeing it. The shape of a strip of sidewalk reflects a deliberate choice — about how pieces fit together to carry weight and drain water. The Appian Way's lava blocks made that same choice two thousand years earlier. The next time you cross a sidewalk, you're stepping on the same idea that carried Roman legions, inspired Escher's fish and lizards, and now helps astronomers map the universe.

Frequently asked questions

What is tessellation in simple terms?

Tessellation is covering a flat surface with repeating shapes so no gaps or overlaps appear. Bathroom tiles, honeycomb cells, and brick walls are everyday examples.

Do astronomers use tessellation to map the sky?

Yes. Sky surveys divide the celestial sphere into gapless tiles so every observation can be filed by location. NASA's Roman Space Telescope uses HEALPix; Pan-STARRS uses a separate grid of projection cells called "skycells."

What shapes can tessellate a flat surface?

Only three regular polygons tessellate a flat plane alone: equilateral triangles, squares, and hexagons. Other shapes can tessellate in combination, or with modified sides — the technique Escher used for his animal tilings.

Is the honeycomb a natural example of tessellation?

Yes. Honeybees build wax cells in a hexagonal pattern with no wasted space. In 1999, mathematician Thomas Hales proved the hexagonal grid uses the least total perimeter of any equal-area partition of a plane — confirming what mathematicians had long suspected but never proved.

How is tessellation different from a mosaic?

Tessellation refers to the mathematical principle: repeating shapes with no gaps. A mosaic is an artistic technique using small pieces of stone, glass, or tile to form an image — often built on a tessellating background, as in the Battle of Issus mosaic.

Keep reading:

• Beatrice Tinsley: the astronomer who changed how we understand the universe
• Hedy Lamarr's patent: what it actually did, and why it took 50 years to matter

Sources & references

• Encyclopaedia Britannica — M. C. Escher biography: britannica.com/biography/M-C-Escher
• American Mathematical Society — "How Did Escher Do It?" (H. S. M. Coxeter's influence on the Circle Limit series): ams.org/publicoutreach/feature-column/fcarc-circle-limit
• Encyclopaedia Britannica — Tessellated pavement: britannica.com/technology/tessellated-pavement
• Encyclopaedia Britannica — Appian Way: britannica.com/topic/Appian-Way
• University of Chicago / Penelope — Appian Way, Britannica 1911 edition: penelope.uchicago.edu
• The Guardian — Appian Way designated UNESCO World Heritage Site, July 28, 2024: theguardian.com
• Thomas C. Hales — "The Honeycomb Conjecture" (1999/2002): arxiv.org/abs/math/9906042
• Wolfram MathWorld — Regular Tessellation: mathworld.wolfram.com/RegularTessellation.html
• The Metropolitan Museum of Art — Primary Characteristics of Islamic Geometric Decoration: metmuseum.org
• Concrete Masonry & Hardscapes Association — Structural Design of Interlocking Concrete Pavement: cmha.org
• ACM SIGGRAPH History Archives — "Computational Fluid Dynamics in a Traditional Animation Environment" (Witting): history.siggraph.org
• Space Telescope Science Institute — Roman Space Telescope, Skymap Tessellation (HEALPix): roman-docs.stsci.edu
• STScI / MAST — Pan-STARRS1 Sky Tessellation Patterns: outerspace.stsci.edu
• The British Museum — MUL.APIN tablet, Library of Ashurbanipal (Late Babylonian astronomical compendium): britishmuseum.org/collection/object/W_Rm-319

This article is for educational and informational purposes only. Sources are linked where available. Readers are encouraged to consult primary sources for further research.