Sacred geometry figures

The Plane 2D · Compass and straightedge

Genesis

• The MonadThe circle and its centre
• Vesica PiscisTwo circles, each through the other’s centre
• Seed of LifeSeven circles, six around one
• Flower of LifeOverlapping circles on a hexagonal lattice
• Fruit of LifeThirteen circles that just touch
• Metatron’s Cube13 centres, 78 lines
• Tree of LifeTen sefirot, twenty-two paths
• Torus YantraCircles turning around a point

Star Polygons

• HexagramStar of David · Seal of Solomon · {6/2}
• Unicursal HexagramSix points in one unbroken line
• PentagramThe golden star {5/2}
• HeptagramThe seven-pointed stars {7/2} and {7/3}
• OctagramStar of Lakshmi · two squares

Ratio & Proportion

• Golden SpiralFibonacci squares and quarter arcs
• PhyllotaxisThe sunflower’s golden angle
• Golden TriangleAnd its gnomon, whirling inward
• Kepler TriangleSides 1 : √φ : φ
• Egyptian TriangleTwelve knots, sides 3 · 4 · 5
• The Sacred CutFour arcs that divide a square

Pattern & Tiling

• Islamic Star PatternHankin’s polygons-in-contact method
• Penrose TilingTwo rhombi, fivefold, never repeating

Yantras & Symbols

• Sri YantraNine interlocking triangles · 43 small ones
• Tetractys1 + 2 + 3 + 4 = 10
• TriquetraThree vesicas · Borromean circles
• EnneagramNine points: the triangle and the hexad
• Lo Shu SquareThe magic square of the river turtle
• Chartres LabyrinthEleven circuits, one path

Harmonic Curves

• Rhodonea Roser = cos(k·θ)
• SpirographHypotrochoid: a circle rolling inside a circle
• Chladni FigureNodal lines of a vibrating square plate

Space 3D · Solids, spheres and flow

Platonic Solids

• Tetrahedron{3,3} · Fire
• Cube{4,3} · Earth
• Octahedron{3,4} · Air
• Icosahedron{3,5} · Water
• Dodecahedron{5,3} · Aether
• Golden RectanglesThree φ rectangles inside the icosahedron

Archimedean Solids

• Truncated Tetrahedron3.6.6 · t{3,3}
• Vector EquilibriumCuboctahedron · 3.4.3.4
• Truncated Cube3.8.8 · t{4,3}
• Truncated Octahedron4.6.6 · t{3,4}
• Rhombicuboctahedron3.4.4.4 · rr{4,3}
• Truncated Cuboctahedron4.6.8 · tr{4,3}
• Snub Cube3.3.3.3.4 · sr{4,3}
• Icosidodecahedron3.5.3.5 · r{5,3}
• Truncated Dodecahedron3.10.10 · t{5,3}
• Truncated IcosahedronThe buckyball · 5.6.6
• Rhombicosidodecahedron3.4.5.4 · rr{5,3}
• Truncated Icosidodecahedron4.6.10 · tr{5,3}
• Snub Dodecahedron3.3.3.3.5 · sr{5,3}

Catalan Solids

• Rhombic DodecahedronThe cell of the densest sphere packing
• Rhombic TriacontahedronThirty golden rhombi

Transformations

• Cube to OctahedronTruncation: cutting the corners away
• Dodecahedron to IcosahedronTruncation through the golden solids
• The JitterbugVector equilibrium → icosahedron → octahedron

Kepler–Poinsot Solids

• Small Stellated Dodecahedron{5/2, 5} · Kepler’s urchin
• Great Dodecahedron{5, 5/2}
• Great Stellated Dodecahedron{5/2, 3}
• Great Icosahedron{3, 5/2}

Compounds

• Star TetrahedronMerkaba · stella octangula
• Five TetrahedraCompound of five tetrahedra / five cubes

Spheres & Flow

• Egg of LifeEight spheres on the corners of a cube
• Sphere Packing13 spheres of the vector equilibrium
• TorusWoven from Villarceau circles
• Torus Knot(p, q) winding
• Fibonacci SphereThe golden angle wrapped around a ball

Cosmos

• Kepler’s Cosmic MysteryMysterium Cosmographicum, 1596

Forms of the Plane

• The SphereThe circle turned through every direction
• Vesica SphaeraTwo spheres, each through the other’s centre
• Flower of Life in 3DSpheres through each other’s centres
• Fruit of Life in 3DTwenty-five spheres on twelve rays
• Metatron’s Cube in 3D25 centres of the 3D Fruit, all joined
• Tree of Life in 3DOne shadow, many bodies
• The Six DirectionsSefer Yetzirah’s sealed space
• Circles and the LineLuria’s igulim and yosher
• Horn TorusSpheres turning around a point
• Golden ShellThe golden spiral given a body
• Chladni CubeSand figures on every face of a vibrating cube
• Clelia CurveThe rose wound onto a sphere
• Spirograph on the TorusThe pen’s loops wound around a ring
• Star Pattern PolyhedronHankin’s method on a closed solid
• Enneagram of the OctaveThe circle rising through a scale
• Icosahedral QuasicrystalTwo golden rhombohedra, never repeating
• Maha MeruThe Sri Yantra raised into a mountain
• Pyramid of the Tetractys1 + 3 + 6 + 10 = 20 spheres
• Borromean RingsThree rings, linked only all together
• Square AntiprismTwo squares turned 45°, joined by eight triangles
• The Kepler PyramidA square pyramid on the Kepler triangle

Hyperspace 4D · Shadows of the fourth dimension

Regular Polytopes

• 5-Cell{3,3,3} · the 4D simplex
• Tesseract{4,3,3} · the hypercube
• 16-Cell{3,3,4} · the 4D cross-polytope
• 24-Cell{3,4,3} · the octaplex
• 600-Cell{3,3,5} · the tetraplex
• 120-Cell{5,3,3} · the dodecaplex

Transformations

• Tesseract to 16-CellTruncation in four dimensions

Hyperspace Flows

• DuoprismPolygon × polygon
• Clifford TorusA flat torus inside the 3-sphere
• Hopf FibrationThe 3-sphere woven from circles

Unfoldings

• Unfolded TesseractThe Dalí cross: eight cubes folding into 4D