05of Eight Branches · The Divinity Rule
The Plastic Number

The
Dimension
Rule.

The proportion that scales across dimensions. Three-dimensional phi.
ρ³ = ρ + 1  ·  ρ ≈ 1.32471795...

The Plastic Number (ρ ≈ 1.3247) is the unique positive real solution to x³ = x + 1. Where phi governs two-dimensional proportion, the Plastic Number extends proportion into three dimensions. Discovered by architect Dom Hans van der Laan, who used it to design his Abbey of Saint Benedictusberg over 30 years.

The Principle Visualized

The mathematics at work.

Core Concepts

What the dimension means.

01
The Plastic Number
ρ ≈ 1.3247
Solves x³ = x + 1. The three-dimensional extension of phi. Where phi relates lengths, rho relates volumes to their sub-volumes.
02
Van der Laan
Abbey of Saint Benedictusberg
Dutch architect Dom Hans van der Laan spent 30 years developing rho as the basis for a complete architectural proportion system.
03
Seven Ratios
ρ&sup0; through ρ&sup6;
Van der Laan identified seven distinct but related ratios the human eye can distinguish. These seven steps form a complete dimensional scale.
04
3D Proportion
Width : Height : Depth = 1 : ρ : ρ²
Objects feel proportionally related when their dimensions relate by rho. The principle for physical product design.
In Practice

How we apply it.

The Dimension Rule governs physical products and three-dimensional brand environments. When a product family needs to feel like it belongs together across three sizes, the Plastic Number provides the mathematical relationship that the eye cannot name but always recognizes.

Product Design
Container Volumes
Bottle heights, product families, and packaging hierarchies sized at rho intervals.
Architecture
Room Proportions
Width : height : depth = 1 : ρ : ρ². Van der Laan's churches use this throughout.
Brand Family
Product Line
Small, medium, large at ρ : ρ² : ρ³. The family reads as related without explanation.
Typography
Three-Level Display Scale
When a display typeface needs three scale levels, rho mediates the steps.
The Full System

The other seven branches.

← Return to The Divinity Rule Root

Build on The Dimension Rule.

License the complete Divinity Rule methodology — all eight branches, full documentation, and application frameworks.

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Branch 05 of 08 · The Dimension Rule · The Plastic Number · The Divinity Rule · Lund Studio LLC · φ = 1.618

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