Studio Matrx Monthly · Volume 1 · Issue 3 · August 2026
Amogh N P
✦ In loving memory of Amogh N P — Architect · Designer · Visionary ✦
An Islamic geometric tile pattern — mathematics as ornament.
Unit IVMathematics in Architecture

Geometry in Architecture

From the golden ratio to tessellations, conics and the catenary.

≈ 35 min · live calculatorByAmogh N P· Architect & interior designer

Geometry is the bridge from number to form. It hides in the proportions of a facade, the pattern of a screen, the curve of an arch and the code of a parametric model. This lesson tours the most beautiful of it — and is honest about the famous myths along the way.

Learning objectives

By the end of this lesson, you will be able to — mapped to the course outcomes for Building Materials & Construction I:

1
CO4 · Understand

Explain the golden ratio and Fibonacci — and judge their architectural claims critically.

2
CO4 · Analyse

Recognise symmetry types and the three regular tessellations.

3
CO4 · Understand

Identify the conic sections and the catenary arch.

4
CO4 · Apply

See how geometry underpins parametric and computational design.

Proportion

The golden ratio — honestly

φ ≈ 1.618 is a real and elegant number, and Le Corbusier really did use it. But the claim that the Parthenon or the Pyramids were “designed on the golden ratio” is historically unsupported — a story from the 1850s. Use φ as a tool, not a law. Try dividing a length in golden proportion below.[1]

The golden ratio & Fibonacci (φ ≈ 1.618) Fibonacci: 1, 1, 2, 3, 5, 8, 13, 21… ratios → φ 8/5 = 1.6, 13/8 = 1.625… Myth alert: no evidence the Parthenon or the Pyramids were “designed on φ” — that story dates only to the 1850s. Le Corbusier's Modulor did use it deliberately — so φ is a real tool, not a universal law of beauty.
DiagramThe Fibonacci sequence, golden rectangle and spiral, with the Parthenon myth flagged
Try it

Divide a length in golden proportion

Larger part (L ÷ φ)
0.0
Smaller part
0.0
Their ratio (φ)
0.000

φ = (1+√5)/2 ≈ 1.618. The larger part relates to the smaller exactly as the whole relates to the larger.

Pattern & form

Symmetry, tessellation & curves

From the symmetry of a plan to the tile patterns of a wall and the perfect curve of an arch — geometry at work. Select a theme.

Symmetry & pattern

Reflection, rotation, translation and glide — the four plane isometries — order facades, domes, screens and the 17 wallpaper patterns.[5]

Only three regular shapes tile the plane Triangle6 × 60° = 360° Square4 × 90° = 360° Hexagon3 × 120° = 360° A regular polygon tiles only if its angle divides 360° exactly — the pentagon (108°) cannot.
DiagramThe only three regular tessellations: triangle, square and hexagon
The catenary — a hanging chain, inverted a chain under its own weight inverted = the ideal arch Hooke: “as hangs the flexible chain, so but inverted stand the rigid arch.” Gaudí used it; the Gateway Arch is a weighted catenary.
DiagramA hanging chain forms a catenary; inverted it is the ideal masonry arch
A tessellated floor — shapes that tile with no gaps.
PhotoA tessellated floor — shapes that tile with no gaps.
A parametric gridshell roof — NURBS geometry built.
PhotoA parametric gridshell roof — NURBS geometry built.
A domed ceiling with rotational symmetry.
PhotoA domed ceiling with rotational symmetry.
A catenary arch — the hanging chain, inverted.
PhotoA catenary arch — the hanging chain, inverted.
Check your understanding

Self-assessment

1. The golden ratio φ is approximately:

2. How many regular polygons tile the plane on their own?

3. The ideal shape for a self-supporting masonry arch is:

In a nutshell

Recap

φ ≈ 1.618 and Fibonacci are real ideas — but the Parthenon 'golden ratio' story is a myth.
Four symmetries (reflection, rotation, translation, glide); only triangle, square and hexagon tessellate regularly.
Conic sections and the hyperbolic paraboloid shape domes, vaults and shells.
An inverted catenary is the ideal masonry arch; geometry now drives parametric, computational design.
The evidence

References & further reading

  1. [1]The golden ratio and Fibonacci, and the myths around them. plus.maths.org. https://plus.maths.org/content/myths-maths-golden-ratio
  2. [5]Symmetry and the 17 wallpaper groups. Wikipedia. https://en.wikipedia.org/wiki/Wallpaper_group
  3. [6]Tessellation, regular tilings and Islamic girih (quasicrystalline) patterns. Lu & Steinhardt, Science 2007. https://peterlu.org/research/islamic_tilings
  4. [7]Conic sections, the catenary and the catenary arch (Hooke, Gaudí, Gateway Arch). Wikipedia. https://en.wikipedia.org/wiki/Catenary_arch
  5. [8]NURBS geometry and parametric design (Rhino / Grasshopper). Parametric Architecture. https://parametric-architecture.com/what-is-rhinoceros-3d-a-3d-modeling-software-based-on-nurbs-geometry/

Further reading

  • Doczi, G. (1981). The Power of Limits: Proportional Harmonies in Nature, Art and Architecture. Boston: Shambhala.
  • Williams, K. (ed.). Nexus Network Journal: Architecture and Mathematics. Basel: Birkhäuser/Springer.
  • Lawlor, R. Sacred Geometry: Philosophy and Practice. London: Thames & Hudson. (read critically — interpretive)

Sources gathered and fact-checked June 2026. Published values vary by source, sample and method — treat as indicative and confirm against the cited standard before structural use.

A

The author

Amogh N P

Architect, interior designer, and creative polymath. Studio Matrx began in his notebooks — his vision of design made honest, useful, and open to everyone. Its Academy is written and taught in his memory, and free, forever.

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