Sacred Geometry Patterns: Fibonacci Spirals, Fractals, Torus & Scale Repetition in Nature
By DailyDestiny Editorial Team
Ever picked up a pinecone or a head of Romanesco and felt it was built on purpose? Learn the five families of sacred geometry patterns, how to count parastichy spirals, measure fractal dimension, and why one pattern repeats from crystals to galaxies.
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What are the five families of sacred geometry patterns?
Imagine you are on a walk and you pick up a pinecone. You turn it over in your hand and notice the scales curling in two directions at once, perfectly neat, as if someone had planned every one. Later, at the market, a head of Romanesco broccoli stops you in your tracks for the same reason.
The short answer: Sacred geometry patterns fall into five families, each defined by the rule that makes it: spiral, tessellation, fractal, proportion and flow. Once you learn the rule instead of the picture, you can read these patterns off a plant, a rock face or a building you have never seen before.
This matters because a pattern is not a symbol. A symbol such as the Flower of Life or Metatron's Cube is a fixed figure with a settled construction and an agreed meaning; those are catalogued in Sacred Geometry Symbols: Complete Guide to 11 Essential Shapes and Their Meanings. A pattern is a procedure, like a recipe, and one recipe can produce endless dishes that look nothing alike. The golden angle is a single rule, and a sunflower disc, a pinecone and an aloe rosette are three of its outputs.
Spiral patterns follow the rule: grow outward while keeping the same shape. They give the logarithmic spiral family and its special case, the golden spiral. You'll notice one when a snail shell seems to grow without ever changing its look.
Tessellation patterns follow the rule: repeat a unit sideways until the surface is covered with no gap and no overlap. They give honeycomb hexagons, triangular lattices, Penrose tilings and Voronoi cell networks. You'll notice one in the cracked mud of a dry puddle.
Fractal patterns follow the rule: apply the operation again to whatever it just produced. They give branching trees, river networks, coastlines and Romanesco broccoli. You'll notice one when a single branch looks like a whole tree.
Proportion patterns follow the rule: fix the ratio between part and whole. They give the golden ratio, the root-2 and root-3 rectangles, and the whole-number ratios of musical harmony.
Flow patterns follow the rule: circulate the field back through its own centre. They give the torus and the vortex. You'll notice one in a smoke ring drifting up from a candle.
You can also identify each family by its invariant, the move that leaves the pattern looking the same.
A spiral is unchanged by rotation combined with scaling: turn a logarithmic spiral through any angle, shrink it by the matching factor, and it lands back on itself. A tessellation is unchanged by translation, since sliding a honeycomb sideways by one cell changes nothing. A fractal is unchanged by scaling alone, so magnifying a Koch snowflake edge by three gives back the original. A proportion pattern survives in ratio when all units are removed. A flow pattern is unchanged under circulation.
Asking what stays the same when you move, spin or zoom a pattern is the most reliable way to identify it in practice, and it is the same logic mathematicians use when they classify patterns by symmetry group.
What is the difference between a sacred geometry pattern and a sacred geometry symbol?
A symbol is a fixed figure; a pattern is a rule that generates figures. The Seed of Life is a symbol: seven circles of equal radius in one settled arrangement, drawn the same way every time, with a stable body of meaning.
Spiral phyllotaxis is a pattern: each new growth point sits 137.5 degrees around from the last. That one rule produces a sunflower, a pinecone and a cactus areole arrangement that look nothing alike. You learn symbols by recognition and patterns by understanding the procedure. Someone who has memorised fifty symbols still cannot say why a pinecone has 8 and 13 spirals; someone who understands the golden angle can predict it.
Can one object display more than one sacred geometry pattern family at once?
Most natural objects show two or three at once, and pulling them apart is a big part of reading patterns accurately.
A pinecone shows the spiral family in its parastichy rows, tessellation in how the scale bases pack the surface without gaps, and proportion in the ratio between successive spiral counts. Romanesco broccoli shows spirals in the arrangement of its buds, fractals in the way each bud is a miniature of the whole head, and tessellation in how the buds pack the surface. A honeycomb shows tessellation in its hexagonal grid and proportion in the constant ratio between cell wall thickness and cell width.
Which sacred geometry pattern family is hardest to verify in a real object?
Proportion, by far. Any complex object has so many measurable distances that some pair will land near 1.618 if you choose the endpoints after the fact. A shell or a facade gives you dozens of candidate landmarks.
Careful practitioners name their measurement endpoints before measuring and report the spread rather than the single best case. Spirals and tessellations are far easier to verify honestly, because parastichy counts and vertex configurations are whole numbers that cannot be nudged.
How do you tell which sacred geometry pattern you are looking at?
When something in front of you catches your eye, you can identify its pattern by asking five questions in order.
First: does the form get wider as it turns around a centre? If yes, it is the spiral family. Confirm it by measuring the radius at two points a quarter turn apart and dividing the larger by the smaller. A constant answer across several quarter turns means a logarithmic spiral rather than a coiled tube of fixed width.
Second: does one unit repeat side by side at the same size? If yes, it is tessellation. Confirm it by counting how many units meet at a vertex: three for hexagons, four for squares, six for equilateral triangles.
Third: does the same shape reappear when you look at a small part of the whole? If yes, it is fractal. Confirm it by finding the resemblance at three or more successive levels, not just one.
Fourth: does the same ratio show up between successive parts, such as consecutive chambers in a shell? If yes, it is the proportion family.
Fifth: does the form loop back through its own middle, so what goes out one side returns through the centre? If yes, it is the flow family.
The order matters. The spiral and tessellation tests give whole-number answers that either match or do not, while the proportion test is the one most easily fooled by wishful measurement. Think of it like checking the facts before you trust your feelings.
How do you measure the growth factor of a spiral in a shell or plant?
Fix the centre point, draw a straight ray outward, and record the distance from centre to spiral where the ray crosses it. Do the same one full turn later and divide the second distance by the first. That is the growth factor per revolution.
Split that figure into four equal multiplicative steps to get the factor per quarter turn, the number usually quoted. A true golden spiral gives 1.618 per quarter turn, about 6.85 per full turn. A chambered nautilus typically gives roughly 3 per full turn, which is logarithmic but not golden. Consistency across several turns matters more than the exact value.
How do you count the spirals in a sunflower head or pinecone?
Put a finger on one seed or scale near the outer edge. Follow the steepest curve clockwise until you return to the starting radius, marking each seed you pass. Then repeat for the counterclockwise set. The two totals are the parastichy numbers.
On a pinecone they are usually 8 and 13, on a pineapple 8, 13 and 21, and on a large sunflower head 34 and 55 or 55 and 89. Marking the first seed with a pin stops the most common mistake, drifting onto a neighbouring spiral partway round. Consecutive Fibonacci numbers in the two directions confirm golden-angle phyllotaxis.
What is the most common mistake when identifying sacred geometry patterns?
Calling something a fractal after seeing one level of resemblance. One fern frond that looks a bit like the whole fern is not proof of a fractal rule; the same shape at three or more successive scales is.
The second mistake is treating any spiral as golden, when most natural spirals are logarithmic with growth factors nowhere near phi, the nautilus included. The third is laying a golden ratio grid over a photo after choosing the crop, which guarantees a fit. Naming the family correctly matters more than finding a famous number, because the family tells you what process made the form.
How does the Fibonacci spiral pattern form in plants?
If you have ever stared at a sunflower and wondered who arranged all those seeds, the answer is a single, quiet rule repeated thousands of times.
It is called phyllotaxis: the order in which a growing tip lays down new organs at a fixed rotation from the last. At the tip of every shoot sits a dome of dividing cells called the apical meristem. Around its rim, new bumps called primordia appear one at a time, each destined to become a leaf, a bract, a scale, a petal or a seed.
Each new primordium forms in the largest available gap, which places it approximately 137.5 degrees around from the one before. That angle, the golden angle, generates the whole pattern. Because 137.5 degrees never divides evenly into 360, no primordium ever lands directly behind an earlier one, so the head fills in without radial gaps or crowded rows. It is like guests filling a round table, each one choosing the emptiest seat.
Here is the surprise: the spirals you see in a sunflower are not laid down as spirals at all. They are an optical result of the sequence. Seeds placed at successive golden-angle steps happen to line up in curved rows, and the number of rows in each direction always comes out as two consecutive Fibonacci numbers.
Small pinecones give 5 and 8, larger pinecones 8 and 13, pineapple surfaces 8, 13 and 21, and big sunflower heads 34 and 55 or 55 and 89. The pattern belongs to the spiral family by its rule, but its Fibonacci counts come from packing, not from any counting mechanism inside the plant. For the ratio itself and its history in art and architecture, see Golden Ratio in Sacred Geometry: 1.618, Greek Temples, Human Body, DNA & Spiral Galaxies.
In the early 1990s, Stephane Douady and Yves Couder showed that the golden angle emerges from physics alone, with no genetic instruction needed. They dropped magnetised fluid droplets one at a time onto the centre of a dish of silicone oil in a magnetic field, so each droplet repelled the others and drifted outward.
When the drops came slowly, they alternated at 180 degrees. As the interval shortened, the arrangement settled on a divergence close to 137.5 degrees with Fibonacci parastichy counts, matching living phyllotaxis. Each new element simply moved to the least crowded spot, and the golden angle is what that requirement produces.
From a sacred geometry standpoint, this deepens the pattern rather than deflating it: the golden proportion is so bound up with optimal spacing that it appears unbidden in a dish of oil with no biology present at all.
Why is the golden angle 137.5 degrees specifically?
The golden angle is 360 degrees divided by phi squared, roughly 360 divided by 2.618, giving approximately 137.508 degrees.
It gives the best spacing because phi is the most irrational of all numbers: the hardest to approximate closely with any simple fraction. Its continued fraction is all ones, which converges more slowly than any other. In practice, rotating by the golden angle takes the longest possible time to come near a previous position. Rotating by 137.5 degrees gives the most even fill, while 120 or 144 degrees, which both divide neatly into 360, produce three or five radial spokes with wasted space between them.
What are parastichy numbers and what do they tell you?
They are the counts of visible spiral rows running clockwise and counterclockwise across a plant surface, and they encode the plant's divergence angle.
A pair of consecutive Fibonacci numbers such as 8 and 13 means a divergence near 137.5 degrees. A pair from the Lucas sequence, such as 4 and 7 or 11 and 18, means a divergence near 99.5 degrees, found in some cacti and sunflowers. Equal numbers mean a whorled rather than spiral arrangement. Reading parastichy numbers is the fastest way to learn a plant's growth rule without measuring any angle. Petal counts follow the same logic, which is why they cluster on 3, 5, 8, 13, 21, 34 and 55.
What makes a pattern a fractal, and how is fractal dimension measured?
You have seen fractals your whole life, in every tree outside your window, and there is a way to put a number on them.
A pattern is a fractal when the same motif reappears at smaller and smaller scales because its rule is applied to its own output. Fractal dimension measures how thoroughly the form fills the space it occupies.
You calculate it by box counting: lay a grid of squares of side length r over the pattern, count the squares that contain any part of it, then shrink r and count again. For patterns built from exact copies, the answer follows straight from the rule:
Koch snowflake edge: each segment becomes four segments one third as long, giving log 4 divided by log 3, about 1.26.
Sierpinski triangle: keeps three half-size copies, giving log 3 over log 2, about 1.585.
Cantor set: keeps two copies one third the size, giving about 0.63, less than a line.
Menger sponge: keeps twenty copies one third the size, giving about 2.73, less than a solid.
Natural fractals give fractional answers too. The coastline of Britain measures around 1.25: more crumpled than a straight line, far from filling a surface. The number tells you how aggressively a form uses its space. That is why lung airways and blood vessel trees, both under pressure to reach every point of a volume, come out very close to 3.
Exact versus statistical self-similarity separates mathematical fractals from natural ones. A Sierpinski triangle is exactly self-similar: every zoom gives an identical copy. A coastline, a cloud edge or a fern is statistically self-similar: zooming gives the same roughness and dimension, but not the same shape.
Natural forms are usually self-affine rather than self-similar, scaling by different amounts horizontally and vertically, which is why a mountain profile looks like a mountain rather than a fern.
Benoit Mandelbrot's 1982 book The Fractal Geometry of Nature argued that Euclid's smooth lines and spheres are the exception in the physical world, not the norm. Sacred geometry had made the qualitative version of that claim for centuries; fractal dimension turned it into a measurement.
Which named mathematical fractals are worth knowing for sacred geometry?
Eight cover most of what you will meet. The Koch snowflake builds an infinite perimeter around a finite area. The Sierpinski triangle removes the central quarter of a triangle again and again, and the same figure appears in Cosmatesque church pavements centuries before it was described mathematically.
The Cantor set removes middle thirds of a line, and the Menger sponge is its three-dimensional cousin, drilling square holes through every scale. The Apollonian gasket packs circles into the gaps between mutually tangent circles forever. The Mandelbrot set comes from iterating z equals z squared plus c in the complex plane. The Barnsley fern is generated by four repeatedly applied affine transformations. The dragon curve emerges from folding a strip of paper in half over and over.
What is an iterated function system and why does it matter for pattern work?
It is a small set of shrink-and-move instructions applied over and over to a starting shape, the simplest machine that makes fractals. The Barnsley fern uses only four, each specifying a shrink factor, a rotation and a shift. Apply them at random hundreds of thousands of times and a detailed fern appears, with stem, fronds and leaflets.
The fern is not stored anywhere; it lives entirely in four short rules. That is why it matters for sacred geometry: it shows what the tradition claims philosophically, that a compact set of relationships unfolds into elaborate structure with no blueprint of the final form existing in advance.
Where do fractal patterns appear in the human body?
Wherever your body has to reach every point of a volume from a single entry. Your bronchial tree divides through roughly twenty three generations from windpipe to alveolar sac, packing an enormous gas-exchange surface into your chest. Your arterial tree branches from aorta to capillary across four orders of magnitude of vessel diameter, using the same bifurcation ratio at each step.
Dendrites branch fractally around each neuron, the folding of the cerebral cortex follows fractal geometry to fit its surface into the skull, and bile ducts and kidney tubules repeat the pattern. Even heart rate variability shows fractal structure, in time rather than space.
Which traditional designs were built on self-similarity before fractal mathematics existed?
Several traditions built them centuries before fractal dimension was defined. The shikhara of a North Indian temple in the Nagara style is clad in miniature copies of the tower's own silhouette, and those carry smaller copies, giving three or four visible levels of the same profile.
Cosmatesque pavement work in Italian churches sets nested triangles inside triangles in what is now called the Sierpinski triangle. The jewel-net of Indra in Buddhist sutra literature places a reflecting jewel at every knot of an infinite net, each holding the reflections of all the others without end. Settlement plans in parts of West and Central Africa repeat one ring layout at the scale of household, compound and village, documented in Ron Eglash's work on African fractals.
Which tessellation patterns fill a surface without gaps?
Every time you walk across a tiled floor, you are standing on a very old question: which shapes fit together perfectly?
Only three regular polygons tile a flat surface on their own with no gaps or overlaps: the equilateral triangle, the square and the hexagon. Every other tessellation uses mixtures of shapes, irregular shapes, or rules that drop periodic repetition altogether.
The hexagon has special standing because it encloses the most area for the least boundary of any shape that tiles the plane, a long-standing conjecture eventually proved by the mathematician Thomas Hales. That efficiency is why honeycomb cells, soap froth walls, convection cells in a heated fluid layer and the cooling joints in basalt formations such as the Giant's Causeway all arrive at hexagons without being related to one another. You'll notice it next time you look closely at bubbles in your sink.
Archimedean tilings. Mixing two or three regular polygons gives the eight semi-regular tilings, each named by the shapes meeting at a vertex, such as the 3.6.3.6 tiling of triangles alternating with hexagons.
Penrose tilings break the rule that a pattern must repeat. Built from two rhombs, or from kite and dart shapes, they cover the plane completely, show local fivefold symmetry, and never repeat by translation however far you extend them.
Voronoi tessellations drop regularity entirely, dividing a surface into the territory nearest each of a set of scattered seed points. That is the pattern of a giraffe's coat, a dragonfly's wing venation, dried mud polygons and the cells of a leaf's epidermis.
Aperiodic tilings turned out to describe real matter. In the early 1980s, Dan Shechtman recorded an electron diffraction pattern from a rapidly cooled aluminium-manganese alloy showing sharp tenfold symmetry. Classical crystallography said that was impossible, because no repeating lattice can carry fivefold or tenfold symmetry. The material was a quasicrystal, ordered but never periodic, with a structure matching the mathematics of a Penrose tiling, and the finding overturned the definition of a crystal.
Islamic geometric ornament reached closely related constructions far earlier through the girih system, where craftsmen assembled star patterns from a standard set of decorated tiles. Researchers Peter Lu and Paul Steinhardt argued that certain late medieval girih designs, notably on the Darb-i Imam shrine in Isfahan, achieve genuinely quasi-periodic order of the Penrose kind.
Why are there only three regular tessellations?
Because the polygon's interior angle has to divide exactly into 360 degrees for copies to close around a point with no gap. An equilateral triangle has 60 degree angles, and six make 360. A square has 90 degree angles, and four make 360. A regular hexagon has 120 degree angles, and three make 360.
A regular pentagon has 108 degree angles: three give 324 with a 36 degree gap, and four overlap. Every regular polygon above six sides has angles over 120 degrees, so only two fit around a point and two can never close a full turn. On a flat surface the limit is absolute, though hexagons and pentagons combine happily on a sphere.
What makes Penrose tilings aperiodic and why does that matter?
The matching rules on their tiles, usually the fat and thin rhomb pair or the kite and dart pair, let the plane be covered completely but forbid any arrangement that repeats by simple sliding. Slide the finished tiling any direction and it never lands back on itself, even though every finite patch recurs elsewhere infinitely often.
Roger Penrose described these tilings in the 1970s. For sacred geometry they resolve an old tension: fivefold symmetry, the geometry of the pentagram and the dodecahedron, cannot form a repeating lattice, yet it can still fill space with perfect order. Order and repetition are not the same thing, and there is something comforting in that.
What are girih tiles in Islamic geometric design?
Girih tiles are a set of five equilateral polygon shapes Islamic craftsmen used to lay out complex star patterns: a regular decagon, an elongated hexagon, a bowtie, a rhombus and a regular pentagon.
Each tile carries drawn line segments, and when tiles sit edge to edge those segments join into the continuous interlaced strapwork you see on the finished wall, while the tile edges themselves disappear. Working from a tile set instead of drawing each line let designers build great complexity without cumulative compass error. The word girih means knot in Persian, naming the points where the strapwork crosses.
Which pattern family does the torus belong to?
If you have ever watched a smoke ring roll through the air, you have seen the one family defined by movement rather than stillness.
The torus belongs to the flow family. Where a spiral describes growth and a tessellation describes coverage, a torus describes a field that leaves a centre, spreads around the outside, returns through the middle and starts again. That is why it is used to model magnetic fields, smoke rings, vortex flow and, in energy work, the field around the body.
Its circulating rule is what sets it apart from the static forms. The shape itself, where it occurs in nature and how it is used in meditation are covered in full in Torus in Sacred Geometry: Doughnut Shape, Earth's Magnetic Field, Heart Energy & Universal Model.
How do sacred geometry patterns repeat at every scale of the universe?
Here is where it can take your breath away: the same few patterns run through roughly forty orders of magnitude, from atomic spacing at around one ten-billionth of a meter to the cosmic web at hundreds of millions of light-years. You can climb the ladder one rung at a time.
Atomic: carbon atoms in graphene sit on a hexagonal lattice with the same vertex configuration as a honeycomb.
Molecular: the DNA double helix is a pair of intertwined helices with a fixed pitch.
Cellular: epidermal cells on a leaf form a Voronoi tessellation, and a neuron's dendrites branch fractally.
Organism: phyllotaxis places seeds by the golden angle, and tree crowns branch fractally.
Geological: cooling basalt cracks into hexagonal columns, and river networks carve fractal drainage patterns with consistent branching ratios.
Planetary: magnetic fields form tori, and storm systems spiral logarithmically.
Galactic: spiral arms trace logarithmic spirals whose pitch angles can be measured directly from images.
Largest mapped scale: galaxies string along filaments and sheets around voids, in a structure statistically self-similar up to about 100 megaparsecs.
The same five families show up at every rung, produced by completely unrelated physical processes each time. Sacred geometry reads that independence of mechanism as evidence of a single ordering principle rather than a pile of coincidences. It is like hearing the same melody played by a violin, a choir and the wind: different instruments, one tune.
Physics has its own name for this. A quantity is scale-invariant when its governing relationship is a power law, meaning there is no characteristic size to anchor it. Power laws describe earthquake magnitudes, city population distributions, the branching ratios of vascular trees and the clustering of galaxies, and any process governed by one looks statistically identical at any magnification.
The renormalization group, developed by Kenneth Wilson and others in the 1970s, was built to explain why systems near a critical point, such as water at the boiling transition, become scale-free and show fluctuations of every size at once.
Sacred geometry does not need the mathematics to make its point, but the overlap is exact: what the tradition calls as above so below, physics calls scale invariance.
What is the difference between scale invariance and self-similarity?
Self-similarity belongs to a shape; scale invariance belongs to a law. A fern is self-similar because a small piece resembles the whole. Earthquake energy release is scale-invariant because the same statistical link between magnitude and frequency holds for minor tremors and major quakes, even though an earthquake has no shape to compare.
Every self-similar shape comes from a scale-invariant rule, but many scale-invariant rules produce nothing you would recognise as a repeating picture. Keeping the two apart prevents a common overreach: assuming that a law described by a power law must look like a fractal image.
Does the large-scale structure of the universe show fractal properties?
Galaxy distribution is statistically fractal up to roughly 100 megaparsecs, about 300 million light-years, and appears homogeneous beyond that. Redshift surveys map galaxies onto a cosmic web of dense clusters joined by filaments and sheets, with enormous near-empty voids between them. The clustering statistics across that range give a fractal dimension near 2: more than a surface, but well short of filling space evenly.
Above the homogeneity scale the pattern stops repeating and the universe looks the same everywhere, as the cosmological principle requires. So the cosmic web is fractal within a bounded range of scales, which is true of every physical fractal.
Which patterns appear at the smallest scales of matter?
Tessellation and polyhedral packing dominate, because atoms in solids settle into repeating lattices. Graphene uses a hexagonal lattice, table salt a cubic one, and close-packed metals stack spheres in arrangements filling about 74 percent of the available volume.
Snowflake crystal habit inherits the hexagonal symmetry of the ice lattice, which is why every snow crystal has six arms. Viral protein shells and radiolarian skeletons build with icosahedral symmetry, covered in Platonic Solids in Sacred Geometry: Tetrahedron, Cube, Octahedron, Dodecahedron & Icosahedron. Quasicrystals sit in the odd corner, ordered like a Penrose tiling and periodic like nothing at all.
Questions people ask
What are the main sacred geometry patterns?
There are five families, each defined by the rule that makes it. Spiral patterns grow outward while keeping their shape, giving logarithmic and golden spirals in shells, plant rosettes and storms. Tessellation patterns repeat a unit sideways until a surface is covered with no gaps, giving hexagonal honeycomb, triangular lattices, Penrose tilings and Voronoi cell networks. Fractal patterns feed a rule back into its own output, giving branching trees, coastlines, Romanesco broccoli and the Mandelbrot set. Proportion patterns hold a fixed ratio between part and whole, the golden ratio being the most cited. Flow patterns circulate a field back through its own centre, the torus being the standard example. Almost every figure you will meet under the name sacred geometry belongs to one of these five or combines two of them.
Why do sacred geometry patterns repeat at different scales?
Because the rules behind them never mention size. A logarithmic spiral is defined by a growth factor per turn, not a diameter, so the same equation fits a 2 centimeter snail and a 100,000 light-year galaxy without changing a term. A rule that splits one channel into two smaller ones works at any starting width. Hexagonal packing minimises boundary length for equal areas whether the cells are soap bubbles or basalt columns. Any process that follows one of these scale-free rules produces the same visible pattern at whatever scale it runs. Sacred geometry reads that independence from size as the signature of a single organising intelligence expressing itself at every level of creation.
What is the most common sacred geometry pattern in nature?
By sheer count, spiral phyllotaxis, because the great majority of flowering plants arrange leaves, bracts, scales and seeds around a stem using a divergence angle close to 137.5 degrees. Hexagonal tessellation is the strongest runner-up, appearing in honeycomb, snowflake crystal habit, basalt columns and insect compound eyes. Fractal branching is close behind, in tree crowns, river drainage networks, lung airways and lightning channels. For a species-by-species catalog, see Sacred Geometry in Nature: Nautilus Shells, Sunflower Spirals, Honeycomb, Snowflakes & DNA.
How are fractals related to sacred geometry?
Fractals are the mathematical form of sacred geometry's oldest claim: the small repeats the large. A fractal is any pattern whose parts resemble the whole at smaller scales, which is exactly what the Hermetic axiom "as above, so below" says about microcosm and macrocosm. Benoit Mandelbrot coined the term fractal in 1975 and gave the property a measurement in fractal dimension. Designers had been building self-similar figures for centuries without the word: the nested miniature towers on a North Indian temple spire, the nested triangles of Cosmatesque church pavements, and the jewel-net of Indra in Buddhist sutra literature, where every jewel reflects every other jewel, including its own reflections, without end.
What is the difference between a Fibonacci spiral and a golden spiral?
A golden spiral is a continuous logarithmic spiral whose radius multiplies by phi, approximately 1.618, for every quarter turn. A Fibonacci spiral is a piecewise approximation of it, built from quarter circles inside squares whose sides follow the Fibonacci sequence 1, 1, 2, 3, 5, 8, 13. The Fibonacci version is made of circular arcs that change curvature abruptly at each square boundary, while the golden spiral curves smoothly everywhere. The first few arcs visibly diverge; by the eighth square the two are almost impossible to tell apart by eye. Most images labelled golden spiral are actually Fibonacci spirals.
What is the difference between a fractal and a tessellation?
A fractal repeats a motif at different sizes; a tessellation repeats it at the same size in different places. Zoom into a fractal and fresh structure keeps appearing, so a Koch snowflake edge looks equally crinkled at any magnification. Zoom into a tessellation and nothing new appears, because a honeycomb is flat hexagons all the way across at one scale. Formally, fractals are unchanged by scaling, tessellations by translation. The Apollonian gasket combines both, packing circles into the gaps between circles and tiling a region with infinitely many sizes.
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Related topics: sacred geometry patterns, fibonacci spiral pattern, fractal sacred geometry, golden ratio pattern, torus pattern, sacred patterns in nature, geometric patterns spiritual, fractal patterns in nature, tessellation sacred geometry
