The idea that certain proportions, the golden ratio, the five perfect solids, the six-petaled flower, are built into the universe, and that drawing them correctly puts you in contact with its architecture.
Documented
The mathematics underneath the mysticism
The documented core is substantial: Euclid's Elements (c. 300 BC) constructs the regular polygons and proves exactly five regular solids exist; Plato's Timaeus (c. 360 BC) assigns four of them to the elements and the dodecahedron to the cosmos; the pentagon's diagonal divides itself in the golden ratio, a fact Greek geometers knew well. Islamic girih tilings achieve quasi-crystalline patterns five centuries before Penrose. Gothic cathedrals were laid out with compass-and-straightedge ad quadratum and ad triangulum systems, recorded in masons' sketchbooks like Villard de Honnecourt's (c. 1230). India's sulba sutras encode altar geometry with exact irrational constructions.
Assessment: All documented, all genuinely impressive. Ancient builders used geometry as their structural and symbolic operating system. What none of these sources call it is sacred geometry: the term is a modern coinage, popularized in the 1980s and 90s, projected backward onto traditions that simply called it geometry.
Stories
The flower that blooms everywhere
The Flower of Life, six circles interlocked into a hexagonal rosette, is genuinely old as a motif: incised on a granite panel at Abydos (date contested), woven into Roman mosaics, drawn by Leonardo in the Codex Atlanticus beside his vesica piscis studies. The folk story makes it a universal master key, the blueprint of creation known to every ancient culture, suppressed by modern science. Its name, though, was coined in the 1990s by the New Age author Drunvalo Melchizedek, whose books supply nearly every claim now repeated about it.
Assessment: The motif's age is documented; the master-key reading is documented publishing, not archaeology. A sixfold rosette is what a compass does when you doodle with it, which is why it appears everywhere circles are drawn: it is the easiest pretty figure in the world to discover independently.
Interesting claims
The pyramid code and the cosmic ratio
The claims: the Great Pyramid encodes the golden ratio and pi in its dimensions; the Parthenon is a golden-ratio temple; DNA, nautilus shells, hurricanes, and spiral galaxies all follow phi; and meditating on these forms, Metatron's Cube especially, tunes the mind to creation's frequency. Measure enough, the argument runs, and coincidence becomes signature.
Assessment: The pyramid's slope produces phi-adjacent numbers only if you pick the convenient edges; the Egyptians' own mathematical papyri record the seked, a simple run-to-rise slope measure that explains the same dimensions with no mysticism. The Parthenon claim requires drawing the rectangle selectively around whichever parts fit. Shells and galaxies follow logarithmic spirals with varying growth rates, only loosely near phi. As with fine-tuning, when any number may count as a hit, everything scores. The wonder is real; the warrant is not.
Sources and further reading
Euclid, Elements (c. 300 BC), Books IV and XIII, the constructions and the five solids
Plato, Timaeus 53c-56c, the elemental solids
Villard de Honnecourt, Sketchbook (c. 1230), the masons' layout methods
Rhind Mathematical Papyrus (c. 1550 BC), the seked slope system in Egypt's own words
Melchizedek, The Ancient Secret of the Flower of Life (1999), the modern canon, cited as the source of the claims
Markowsky, Misconceptions about the Golden Ratio, College Mathematics Journal (1992), the measurement audit
Livio, The Golden Ratio (2002), the honest history
Worth asking
When a ratio can be found in a pyramid, a pineapple, and a galaxy, is the universe speaking geometry, or is the measurer just very motivated?