The Geometry
Geometry Primer
An interactive introduction to the fundamental geometric concepts, number systems, and methods used by Palladio — designed to equip the reader to understand, weigh, and evaluate the evidence that follows.
The Equilateral Triangle
The equilateral triangle is the foundational geometric form in Palladio's design method. Its key property for architectural design is the relationship between its base and its height: if the base has length 1, the height is √3/2 (approximately 0.8660).
This ratio — 1 : √3/2 — appears throughout Palladio's work, encoded in the numerical dimensions he provides in the Quattro Libri. Use the explorer below to see how changing the base affects all other measurements.
Equilateral Triangle Explorer
Base: 100
Height: 86.6025
Side: 100 (equilateral)
Perimeter: 300.00
Area: 4330.13
Ratio 1:√3/2: 1 : 0.866025
Palladio's Number Systems
Since √3/2 is irrational, Palladio used rational number approximations to represent it in his dimensions. These approximations form consistent systems that can be identified across his work. Common approximations include 7:6, 15:13, and 26:22½.
Understanding these number systems is the key to decoding Palladio's marked and unmarked dimensions. Enter any base dimension below to find the corresponding height and its nearest whole-number approximations.
Ratio Calculator
Enter a base dimension to see the corresponding height using Palladio's equilateral triangle ratio (1 : √3/2). Palladio used whole-number approximations of these irrational values.
Exact height: 25.9808
Nearest whole number: 26(error: 0.07%)
Nearest half: 26.0(error: 0.07%)
Common approximations
| Base | Exact h | Approx | Error |
|---|---|---|---|
| 2 | 1.73 | √3 | 0.0% |
| 4 | 3.46 | 3½ | 1.0% |
| 6 | 5.20 | 5.196 | -0.0% |
| 7 | 6.06 | 6 | -1.0% |
| 8 | 6.93 | 7 | 1.0% |
| 10 | 8.66 | 8⅔ | -0.0% |
| 12 | 10.39 | 10.39 | -0.0% |
| 14 | 12.12 | 12 | -1.0% |
| 15 | 12.99 | 13 | 0.1% |
| 20 | 17.32 | 17⅓ | -0.0% |
| 24 | 20.78 | 20.78 | -0.0% |
| 26 | 22.52 | 22½ | 0.0% |
| 28 | 24.25 | 24 | -1.0% |
| 30 | 25.98 | 26 | -0.0% |
Proportional Overlays
Palladio used equilateral triangle templates to generate proportional frameworks for his designs, both in plan and in elevation. A single triangle establishes the basic height-to-width relationship. Paired or “starred” triangles create a richer set of intersection points that correspond to key architectural features.
Experiment with different overlay modes below to see how these geometric templates generate subdivision grids. The “double” mode shows the six-pointed star (hexagram) formed by two overlapping equilateral triangles — a figure central to Palladio's method.
Proportional Overlay
Visualize how equilateral triangle overlays generate the proportional frameworks Palladio used in plan and elevation design.
Moving Beyond the Cartesian Grid
Previous analyses of Palladio's proportions — most notably Wittkower's harmonic ratios — used the framework of a two-dimensional Cartesian grid. This approach obscured relationships that become obvious once we recognise the equilateral triangle as the generative form.
The dissertation demonstrates that these geometric methods were not unique to Palladio. The use of geometric templates based on the equilateral triangle was common practice among pre-modern architects, traceable from ancient Roman practice through medieval masonry craft to the Renaissance.
Armed with the tools on this page, readers can examine buildings from any pre-modern period for the possible presence of these geometries and methods.