## Structure with Rings of Proportionally Identical Arched Vaults

2002 Arches

### Aesthetic & Mathematical Inspiration

#### Escher and Coxeter

M. C. Escher responded to the some of the mathematician H. S. M. Coxeter’s work. Specifically, Escher was fascinated with a figure which “depicts a tessellation of the hyperbolic plane by right triangles” Coxeter Tessellation (Wikipedia). The Circle Limit III, 1959 was a result of M. C. Escher’s ‘Limit’ tessellations. Another is the print Square Limit, 1964. It is in the National Gallery of Art. In both the upper size ‘Limit’ is obvious. Both Escher’s and Coxeter’s creations of limit tessellations of the plane are two dimensional. I was aware of some of Escher’s ‘Limit’ pieces, but not Coxeter’s. In conclusion, my arched form displays how this hyperbolic limit approach can be applied three dimensionally in real world architecture

#### An Apt Comparison

Architecturally the space created is suggestively symbolic. As a point of reference I refer you to the Museum of Anthropology, University of British Columbia, Vancouver, BC. Upon entry one is in a relatively narrow, low ceilinged hallway which stretches ahead sloping gently down. On the sides are large horizontal wood sculpture of various First Nations, i.e. Haida, etc. Walking ahead one notices the hall is gradually widening and the ceiling is lifting. At the same time now the huge wooden artifacts are now of upright proportions. The crescendo of this walk and its architectural space leads to a much wider, higher space. This space is defined with a glass exterior wall opening to a sculptural garden replete with First Nations’ house frontal (totem) poles. It overlooks the Pacific Strait of Georgia. The experience of moving through the museum is very dramatic. The widening hall with its rising ceiling culminating in the glass curtain and views of sculpted poles, trees and ocean water. This gives it a sense of greatness and unlimited space.

#### Central Simplicity

Walking into this arched structure has a comparable effect although one travels from the outside to the building center. Assuming all 188 (exterior) arched entries are open or have opening doors, almost 200 people could enter from unique entries (and perspectives). As a person moves inward the height and cross-sections of each ring of the arches and their supporting columns increase by the √2. Thus the central arches are 16 times higher and thicker than the smallest, outermost ones. This great innermost scale and simplicity contrasts with the busy, outermost complexity. Meanwhile the very columns and arches of the outer ring precisely repeat and reflect the central ones.

### Side & Plan View

Plan View

The major column springing has pairs of primary (1), secondary (2), tertiary (3), and quaternary (4) arch sizes in the order 12344321.

Side View

The major column springing has pairs of primary (1), secondary (2), tertiary (3), and quaternary (4) arch sizes in the order 12344321.

### The Two Types of Arch Springings and Plan View

Major & Minor

Major Column Springing

The major column springing has pairs of primary (1), secondary (2), tertiary (3), and quaternary (4) arch sizes in the order 12344321.

Plan view of a proportional arched structure

Each vault is a regular 45° 90° 45° right triangle in the plan view. The result is arches whose sizes are related by the square root of 2. This plan view has nine ‘rings’ of vaults. Additional ‘rings’ can be added until their size is impractically small. The intermediate column/arch/vault sizes are 1/√2 smaller than the next size larger column/arch/vault. It is also √2 larger than the next smaller column/arch/vault. A structural unit includes the arch, its column parts and its vault. Each column is the meeting of eight column parts. Is the ratio of the two types of springings a limit in Coxeter’s sense?

Minor Column Springing

The minor column springing has pairs of secondary (2), tertiary (3), and quaternary (4) arch sizes in the order 22344322.

### Poster and Perspective View

Poster of cutaway view of proportional arch structure surrounded with a pattern of other views

View from ground level into and across the proportional arch structure center

### Views of Ceiling with Arches & Vaults

View of vaults

View of Major Vault Springing