Frontier of the Finite
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Images and diagrams from the book and more
From the Book
Seventeenth century Siberian shaman
The Greek God, Pan
Direct Realism
The Explanatory Gap
Plato's Cave
The sphere passing through Lineland
A Flatland House
Optical Illusions: Checkerboard, Necker Cube and Penrose Triangle
All aspects of reality are within our consciousness
World of the Real
Seventeenth-century woodcut of witches
Descartes's diagram of the pineal gland
The Platonic Solids
The Vesica Piscis
The third stage and triquetra
The fourth and fifth stages added
The sixth and seventh stages, the latter creating the seed of life
Seed of Life, torus, seed ratcheted
The Egg of Life in 2D and 3D
The flower of life with 19 circles encircled
Expanded Flower of Life with 61 circles and the Fruit of Life
Metatron's Cube
The Golden Ratio
The Fibonacci Rectangle and Golden Spiral
Gell-Mann's Octet and Decuplet
The Lorenz Strange Attractor
Chaos Bifurcation diagram
The Koch Snowflake, 0–4 iterations
Fractal Weeds
The Menger Sponge over 5 iterations
A Julia Set
The Mandelbrot Set
The Buddhabrot Set
The Bifurcation diagram and Mandelbrot relation
Chladni Plates
Faraday patterns
Cymaglyph
The Torus
The cave of consciousness
Torus diagram
Torus direction
Frontier diagram
Table of consciousness levels
Cymatics
Cymatics: geometric patterns formed by sound frequencies
The Eidophone: voice figures created by singing into a vibrating membrane
The Eidophone: voice figures created by singing into a vibrating membrane
Cymatics: a 3835 Hz Chladni pattern mirroring the ancient Babylonian star symbol of the sun god Shamash
Cymatics: a Chladni vibration pattern mirroring the geometry of a tortoise shell
Fractals
A family of Julia set biomorphs: organic forms generated by different complex number formulas
Eleven Julia sets illustrating the vast variety of forms produced by varying a single complex parameter
A biomorph with similar structure to a human brain MRI
The Mandelbrot set zoomed in four stages, revealing self-similar structure at every scale
A second Mandelbrot set zoom sequence, showing infinite complexity emerging from a simple equation
DNA nanotechnology: self-assembled molecules forming a Sierpiński triangle across three levels of scale
The fractal branching of the human lungs: the same pattern repeated at ever-smaller scales
A river delta viewed from above, showing fractal branching patterns identical to those in the lungs
Cracked dry earth forming self-similar fractal patterns across its surface
The fractal vein network of a leaf: nature's efficient distribution system repeated at every scale
Romanesco broccoli: a naturally occurring fractal whose spiralling cones repeat the same form at every scale
Geometry in Nature
Fibonacci spirals in a sunflower and pineapple: nature's most efficient packing arrangement
X-ray diffraction pattern revealing the hidden geometric order within a quasicrystal
Electron microscope images of quasicrystal structures displaying icosahedral and dodecahedral geometry
The E8 root system: the most complex geometric symmetry structure known, underlying certain theories of fundamental physics
The geometric pattern traced by Earth and Venus over 8 orbits, forming a near-perfect rose curve
A simplified visualisation of the E8 symmetry group projected into two dimensions
Fibonacci spirals in a pine cone: 8 spirals in one direction, 13 in the other — consecutive Fibonacci numbers
The geometry of the human body: sacred proportions and Fibonacci ratios mapped onto human anatomy
A snowflake crystal displaying perfect sixfold symmetry — the same geometry as the flower of life
Cross-section of a nautilus shell revealing the golden spiral: nature's most elegant growth pattern
Golden ratio proportions mapped onto three fish species: Coho Salmon, Curlfin Sole and Pacific Pompano
Torus Geometry in Nature
The Vector Equilibrium: the Cuboctahedron as a zero-point form underlying toroidal and sacred geometry
Electron density maps of molecular orbitals, revealing toroidal and dumbbell-shaped probability fields
Atomic orbital geometries: the dumbbell (p-orbital) and doughnut (toroidal) shapes of electron probability distributions
The toroidal electromagnetic field of the human heart, extending far beyond the body (Institute of HeartMath)
Hydrogen atom wave function probability density plots: the geometric forms that electrons inhabit at different energy levels
A supermassive black hole surrounded by a dusty torus, ejecting plasma jets along its polar axis (©Nature)
The unit circle: the mathematical foundation of toroidal and wave geometry, mapping all trigonometric relationships
Toroidal forms expressed through vortex mathematics, plasma fields, geometric modelling and harmonic resonance patterns
A tree within a toroidal field: roots and branches as mirror images of the same recursive, self-sustaining form
The Milky Way galaxy: a vast toroidal spiral viewed from above, its arms following the golden ratio
Birth Defects Correlating to Past Life Memories of Children
A child born with missing fingers and toes corresponding to wounds described in a previous life memory
A birthmark on the back of a child's head corresponding to a reported gunshot entry wound in a past life
A birthmark on the side of a child's head corresponding to a reported gunshot exit wound in a past life
A child born with a malformed foot corresponding to an injury described in a previous life memory
A child born with truncated fingers corresponding precisely to wounds described in memories of a previous life