Golden Ratio Construction
History
The golden rectangle and its spiral have been studied since Euclid's Elements (c. 300 BCE), where it appears as the "extreme and mean ratio." The rectangle's unique property — removing a square leaves a smaller rectangle of identical proportions — produces the logarithmic spiral found in nautilus shells, hurricane formations, and spiral galaxies. Renaissance architects from Alberti to Palladio used the ratio to proportion façades; Le Corbusier built an entire measurement system (the Modulor) around it.
Why this section
The opening construction. The golden rectangle subdivides as the eye moves inward, with the spiral drawing the viewer's gaze toward the centre — mirroring how the page itself funnels attention from the brand mark down to the headline.
Flower of Life
History
One of the oldest geometric patterns known, the Flower of Life appears carved into the Temple of Osiris at Abydos (Egypt, c. 535 BCE), in Phoenician art, on medieval church floors, and in Leonardo da Vinci's notebooks. It is constructed by the simplest possible rule: draw a circle, place the compass on its edge, draw another of the same radius, and repeat. Every intersection seeds the next circle.
Why this section
Growth from a single seed — one method, iterated. The About section introduces Catena's design thinking framework: five phases, each built on the one before. The Flower of Life is that idea drawn: one circle generates six, six generate twelve, and the pattern extends as far as you care to take it.
Metatron’s Cube
History
Named after the archangel Metatron in medieval Jewish mysticism, this figure connects the thirteen circles of the Fruit of Life with straight lines through every centre. The resulting mesh contains flat projections of all five Platonic solids — tetrahedron, cube, octahedron, dodecahedron, and icosahedron — the only regular polyhedra that exist.
Why this section
The IB section maps criteria to curriculum. Metatron's Cube is structure made visible — every point connected to every other, a systematic grid behind apparent complexity. The IB rubric is the same: four criteria that look independent but interlock into one design process.
Fibonacci Spiral
History
Leonardo of Pisa (Fibonacci) introduced the sequence 1, 1, 2, 3, 5, 8, 13… to Europe in 1202, though Indian mathematicians had described it centuries earlier. Tiling squares whose sides follow the sequence and connecting their corners with quarter-circle arcs produces a spiral that approximates the golden logarithmic spiral. The pattern appears in sunflower seed heads, pinecone spirals, and the breeding mathematics of rabbits (Fibonacci's original problem).
Why this section
Scaling from a small base. Ontario's 15,000 DECA members compete against a field twenty times larger — and win disproportionately. The Fibonacci spiral is growth that compounds: each generation builds on all the previous ones, and the curve steepens with every turn.
Golden Triangle
History
An isosceles triangle whose base angles measure 72° and whose apex measures 36°. The ratio of leg to base is φ. Bisecting a base angle produces a smaller golden triangle inside the original — and the process repeats infinitely, each subdivision yielding another self-similar triangle and a golden gnomon. The construction appears in the geometry of the regular pentagon, studied extensively by the Pythagoreans.
Why this section
Five sessions, each a self-similar subdivision of the whole process. Frame, Ideate, Build, Refine, Critique. The golden triangle's recursive subdivision — where every cut produces two shapes that contain the ratio again — mirrors a curriculum where each session is complete in itself but gains meaning from its position in the sequence.
Root Rectangles
History
The root rectangles (√2, √3, √5…) are constructed with compass and straightedge from a unit square: swing the diagonal to generate √2, swing the diagonal of the √2 rectangle to generate √3, and so on. Jay Hambidge's "Dynamic Symmetry" (1920) demonstrated their use in Greek vase painting and temple plans. The √2 rectangle is the basis of the ISO paper system — fold A4 in half and the proportions remain identical.
Why this section
The practitioner's tools. Root rectangles are the geometry of classical proportion — learned through practice, not theory. Twenty years of applying these proportions to real products is the credential the Instructor section presents: the rectangles are the work itself, rendered as construction.
Vesica Piscis
History
Two circles of equal radius, each passing through the other's centre. The almond-shaped intersection (the mandorla) has a height-to-width ratio of √3. The figure appears in Romanesque and Gothic architecture as the frame for Christ in Majesty; it is the geometric basis of the pointed arch, the ichthys symbol, and the Seal of Solomon. In optics, the same construction generates Venn diagrams.
Why this section
Two circles meeting. The contact form is where the visitor's world and Catena's overlap — the mandorla is the space of shared interest. One of the simplest constructions in sacred geometry, and the last the visitor sees: an invitation, not a proof.