Geometry is best understood visually. The diagrams below are mathematically precise — every circle, intersection, and ratio is calculated exactly.
Vesica Piscis
The Vesica Piscis is formed when two circles of equal radius overlap so that each centre lies on the other's circumference. This shape is the foundation of sacred geometry — from it, the square root of 2, 3, and 5 can all be derived.
The vertical line through the intersection points has a length of $\sqrt{3}$ times the radius — one of the most fundamental ratios in nature.
Seed of Life
Seven circles of equal radius, each centred on the intersection points of the others, create the Seed of Life — a pattern found in ancient temples from Egypt to China.
Circle Inscribed in Square
A fundamental relationship: the ratio of the inscribed circle's area to the square's area is always $\frac{\pi}{4}$ — approximately 0.785.
Golden Ratio Rectangle
The golden ratio $\phi = \frac{1 + \sqrt{5}}{2} \approx 1.618$ appears throughout nature. This construction shows how to derive a golden rectangle from a square.
Flower of Life
Extending the Seed of Life pattern outward produces the Flower of Life — 19 overlapping circles forming the basis of sacred geometric construction.
Metatron's Cube
Connecting the centres of the 13 circles in the Fruit of Life creates Metatron's Cube — a sacred geometry figure encoding all five Platonic solids within its structure.
Pentagram and Golden Ratio
The pentagram contains the golden ratio in every proportion — each diagonal segment divided by the next smaller creates $\phi$. This is why the five-pointed star has been a symbol of harmony since antiquity.
