An exhibition in two halls · the number & the form

The Golden Ratio

Onward growth from within — the perfect division of unity, so that however diverse the whole becomes, every part answers the same way back to its source.

Every measurement on this page is computed live from the geometry
Descend
Hall II · The Form

From the number to space

A ratio relates two lengths. Ask the same relation to fill a plane and it becomes an angle; ask it to fill a volume and it becomes a solid. Three instruments follow the golden ratio out of the line and into the world.

Instrument · 01

The sunflower engine

The golden cut can also divide a circle. Its smaller angle is 360°/φ² = 137.5077…°, the golden angle. Angles near this value appear in many arrangements of leaves and florets. The study of plant arrangement is called phyllotaxis; the instrument below explores a geometric model of these patterns.

α = 360° / φ² = 137.507 764…° The golden angle · an irrational fraction of a turn

An irrational fraction of a turn never repeats exactly. The golden angle is especially resistant to close approximations by fractions with small denominators, a property connected with φ's continued fraction [1; 1, 1, 1, …]. In this model, successive points advance by a fixed angle while their distance from the center increases with the square root of their number. Rational turns produce spokes; other turns produce different spiral arrangements.

Change the angle to compare those arrangements. At the golden setting, counter-winding spiral families often have consecutive Fibonacci counts. Such counts are common in observed sunflower heads, with other patterns also recorded. The disc, dome, pinecone and sphere offer four views of a shared construction.

INSTRUMENT 01 Phyllotaxis field nudge the angle · drag to orbit · light the spirals
Form Floret Parastichies
Divergence 137.508° ≈ α
Luminance Chroma roll Rotation Seeds Disc spacing index —
Choose a form and angle, then reveal the counter-winding spiral families with the parastichy controls. The spacing index compares average nearest-neighbor distances in a sample of up to 700 outer disc points with a triangular-lattice reference. All four views share this disc measurement. Changing the angle replays the growth.
Instrument · 02

The dodecahedral cascade

The dodecahedron carries fivefold geometry into space. Each of its twelve pentagonal faces contains a pentagram with golden-section crossings. Its dual, the icosahedron, exchanges the roles of faces and vertices.

In this instrument, an icosahedron's vertices meet the face centers of a surrounding dodecahedron. The whole pair then repeats at a chosen scale of 1/φ². The slow inward journey makes that repeated relation visible as a cascade of shells.

INSTRUMENT 02 Nested dodeca / icosa cascade drag to orbit · zoom cascade
Shell depth each pair = previous ÷ φ²
Gold: dodecahedron. Violet: an icosahedron with vertices at its face centers. Each pair repeats at 1/φ² of the previous pair; drag to orbit or adjust the shell depth.
Instrument · 03

Three golden rectangles, at right angles

Three golden rectangles reveal the icosahedron's structure. Center them in mutually perpendicular planes, with corners (0, ±1, ±φ), (±1, ±φ, 0) and (±φ, 0, ±1). Their twelve corners are exactly the solid's twelve vertices. Join each vertex to its five nearest neighbors and thirty edges enclose twenty triangular faces.

Each rectangle has sides 2 and 2φ, so its proportion is 1:φ. The dodecahedron is the dual obtained through the icosahedron's face centers. Together, the instruments show two complementary ways that fivefold geometry enters a solid.

INSTRUMENT 03 Icosahedron from three golden rectangles assemble · orbit · reveal twelve stars
Assembly
Pentagrams
Gold card: the yz-rectangle (0, ±1, ±φ), normal to the gold x-axis. Violet: the xy-rectangle (±1, ±φ, 0), normal to z. Cyan: the xz-rectangle (±φ, 0, ±1), normal to y. Every edge of the finished solid has length 2 — the short side of each card — and every vertex is a corner of exactly one card.

Each vertex has five neighbors in a flat pentagonal ring. Join alternate corners and a pentagram appears: twelve stars in twelve planes, each with golden-section crossings. “Cycle stars” traces, holds and fades each star in turn; “All 12 stars” reveals the full matrix. The plane toggle shows the pentagonal slices through the vertex set. The solid’s twenty faces remain triangular.

Interlude · The Star

The pentagram, before the rumours

The star now moves from construction to interpretation. Its proportions remain fixed while its meanings develop through particular traditions and times.

The pentagram has served as a sign of health in the Pythagorean tradition, a pattern of Christian virtues on Sir Gawain's shield, and an image of the human microcosm in Renaissance occult philosophy. Éliphas Lévi's nineteenth-century writings gave its two orientations contrasting occult readings. The historical graphics below show how the same five points frame an upright human correspondence or an inverted goat head.

INTERLUDE The star, upright and inverted switch the symbol · compare the geometry
Turn
Choose an orientation to switch the historical graphic, or turn the slider to watch one reading become the other. The luminous lines and crossing markers are the measured geometry; the engraving is a historical illustration laid over it. Switch the overlay off to see the construction alone, or switch the microcosm on to compare the human figure. The ideal star keeps its golden ratio through every turn.

Tetragrammaton · the upright microcosm

Lévi's diagram surrounds a point-up pentagram with divine names, planetary signs and ritual emblems. Its upper point represents spirit governing the four elements; the head, hands and feet supply its human correspondence.

Tetragrammaton means the four-letter Hebrew divine name, YHWH; here it names this esoteric diagram. Its progressive letter calculation opens a further numerical reading below. Related forms appear in Samael Aun Weor's modern Gnostic teachings.

Éliphas Lévi · 1855 · public-domain artwork · View the historical source

NAME · NUMBER · ANGLE Four letters, a sum of 72 gematria beside geometry

In Hebrew gematria, letters carry numerical values. YHWH, the four-letter divine name, has the values 10, 5, 6 and 5. Its progressive forms gather those letters into a cumulative pattern.

A progressive form of the name

Y = 10 · H = 5 · W = 6 · H = 5

Progressive forms of YHWH have the combined value seventy-two Y equals ten. Y H equals fifteen. Y H W equals twenty-one. Y H W H equals twenty-six. Adding the four rows gives seventy-two. The four letter positions contribute forty, fifteen, twelve and five, also totaling seventy-two. ROWSUM 1Y1010 2Y10H515 3Y10H5W621 4Y10H5W6H526 401512572 POSITIONAL TOTALSTOTAL
10 + 15 + 21 + 26 = 72The full name alone totals 26. The triangle contains ten letter occurrences; the positional totals regroup the same values.

The angles of fivefold order

36° · 72° · 108°

The regular pentagram contains thirty-six, seventy-two and one hundred and eight degree anglesA highlighted tip triangle has a thirty-six degree apex and two seventy-two degree base angles. The central pentagon has one hundred and eight degree interior angles. Its diagonal crossings pair seventy-two with one hundred and eight degrees.36°72°108°72° = 360° / 536° = 72° / 2 · 108° = 180° − 72°
36° at the tips · 72° at the tip basesThe central pentagon has 108° interior angles. At each diagonal crossing, 72° and 108° meet as supplementary angles.

The gallery places the cumulative value 72 beside the star’s 72° as a symbolic correspondence. The letter calculation follows gematria; the angles follow the geometry of a regular pentagram.

Explore the 36°–72°–72° golden triangle →
Historical context and transliteration

The triangular divine name appears in Agrippa’s Occult Philosophy, II.13, within a Renaissance Christian Kabbalistic treatment of the number ten. This setting brings Hebrew letter symbolism into a triangular form associated with the Pythagorean tetractys, the arrangement of 1 + 2 + 3 + 4 positions. The numerical labels here explain the calculation in the supplied reference diagram.

The transliteration YHVH uses V for the same Hebrew letter ו represented by W in YHWH; its numerical value remains 6. The letter sequence is yod–he–waw–he. In the cited teaching associated with Isaac Luria, this progressive sequence is called the name’s “square form.” Here it is displayed in triangular rows.

Sub-exhibit · Nature, the adversary and the horned image

Pan: the primal pulse of nature

The horned figure opens a history of nature, belief and reinterpretation: from the Greek countryside to later images of the Devil.

Pan's human–goat body joins shepherds and flocks with rustic music, desire and wild places. The gallery encounters him as the primal pulse of nature: life’s generative energy, exhilarating and untamed.

Original painterly interpretation of Pan, a horned adult with human torso and shaggy goat legs, playing reed pipes in a sunlit Arcadian woodland.
The living woodlandMusic, animal life and abundant growth gather around Pan. The image imagines the natural vitality associated with the ancient god. Original AI-generated artwork · contemporary interpretation, 2026 · inspired by classical Pan sculpture and the woodland painting tradition. Historical sculpture · Painting reference: Poussin, 1636
The ancient landscape

Fecundity, music and the untamed

In the Homeric Hymn to Pan, mountains and wooded glades resound with reed music and dancing nymphs. Roman sculpture preserves his horns, beard and shaggy legs, alongside imagery of fertility and abundance.

Fecundity is the capacity to produce and renew life. Pan's animal body, desire and woodland setting give that abundance a vivid presence. His wildness also holds fear and exhilaration together: nature can nourish, overwhelm and unsettle.

Historical anchors: The Met’s Roman Pan or satyr with fertility offerings · Ronald Hutton on Pan’s changing reception.

Language, belief and reinterpretation

How the histories meet

Greek Pan and the Hebrew adversary begin in different religious worlds. Their later encounters belong to a history of changing images and beliefs.

Hebrew scripturesWhat “adversary” means

Hebrew śāṭān means adversary or opponent; “accuser” describes its role in some passages. In Job 1–2, ha-satan challenges Job's motives within God's permission. Later traditions developed the figure in different directions, including the Christian cosmic opponent.

The later term Satanism grew into a name for accusations, ideas and religious identities. Its history continues beyond the Hebrew word.

Job 1:6 and the NET Bible translation notes · Joseph P. Laycock, Satanism.

Changing traditionsHow the Devil acquired familiar features

Christian art gave the Devil many animal and monstrous forms. Historian Ronald Hutton argues that the nineteenth-century revival of Pan helped make goat horns, beard and hooves familiar features of the modern Devil.

This was a later artistic encounter between traditions. Pan belongs to Greek religion; the wider category of paganism includes many other deities and beliefs.

Ronald Hutton, “The Return of the Horned God,” Gresham College.

1854–56 · 1897Baphomet and the inverted star

Lévi's Baphomet appeared in publications of 1854–56. He explicitly associated the figure with Pan as part of his nineteenth-century occult synthesis. The engraving below shows how he assembled its human and animal features.

The 1897 goat head within an inverted star is a separate design, later adapted by the Church of Satan.

Julian Strube, “The ‘Baphomet’ of Eliphas Lévi: Its Meaning and Historical Context” · Church of Satan’s account of its emblem.

Modern identitiesFrom accusation to self-description

Satanism first operated as an accusation and later became a self-description for varied literary and religious positions. Those uses require their own contexts: accusing someone of devil worship says little about what that person actually believed.

The Church of Satan, founded in 1966, describes its religion as atheistic and Satan as a symbol. The separate Satanic Temple also rejects a supernatural Satan. Other self-identified Satanists hold different beliefs, while contemporary Pagan religions have their own histories.

Laycock’s historical overview · Church of Satan · The Satanic Temple FAQ.

Original allegory of a classical Pan figure with reed pipes in warm woodland light beside an enlarged horned shadow on a dark surface.
A form and its shadowSimilar features can invite different stories. This allegory reflects on the viewer’s interpretation of the horned figure. Original AI-generated artwork · contemporary allegory, 2026 · inspired by classical Pan imagery. The shadow is a visual metaphor for reinterpretation, not a depiction of a documented historical event.
Historical image · gesture and interpretation

Baphomet: the hands and the star

Lévi's Baphomet and Christ's blessing hand offer two images of gesture. Their own explanations come first; a modern Gnostic reading and the gallery's comparison follow.

Éliphas Lévi's Baphomet: a winged goat-headed figure with an upright forehead pentagram, two fingers of one hand pointing upward and two of the other downward.
The Sabbatic GoatÉliphas Lévi's Baphomet, reproduced from Dogme et rituel de la haute magie, volume II, 1856. Historical engraving · public domain · Wikimedia Commons source and credit · select the image to view it in full.
Lévi's own explanation

Above, below and equilibrium

Two fingers of one hand point upward; two of the other point downward. In his explanation, Lévi connects the upward gesture with the white moon of Chesed and the downward gesture with the black moon of Geburah: mercy and justice held in equilibrium. The forehead pentagram is upright, which he identifies with light. The whole figure combines opposites within his occult system.

Lévi, Transcendental Magic, translated by A. E. Waite, pp. 289–290.

Sacred Heart devotional image of Christ, with the thumb, index and middle fingers of his blessing hand extended and the ring and little fingers folded.
Christ giving a blessing The thumb, index and middle fingers extend; the ring and little fingers fold inward. Sacred Heart devotional image · source image from the Christianity Stack Exchange discussion.
Glorian · a modern Gnostic reading

The blessing hand and its shadow

In “Alchemical Symbolism of the Nativity of Christ,” Samael Aun Weor reads the blessing hand's three extended digits as Father, Son and Holy Spirit. Its devil-like shadow becomes an analogy for Lucifer as the shadow of the inner Logos, in a Gnostic teaching about light and reflection.

Glorian's separate teaching on the microcosmic star associates one point up with Christ and two points up with the goat of the Witches' Sabbath.

Gallery interpretation · five digits, two readings

A hand can echo a star

Bringing these readings together, the gallery compares five digits with five star points. Two extended and three folded evoke the inverted witch's foot; three extended and two folded evoke the upright, divine star. This is the exhibit's symbolic analogy between gesture and orientation.

Two extended · three folded
The inverted echo

Two star points above, three below.
The gallery's "witch's foot" comparison.

Three extended · two folded
The upright echo

Three star points above, two below.
The gallery's divine-star comparison.

Sources and artwork notes
  1. Éliphas Lévi, Baphomet, 1856, Wikimedia Commons. Public-domain historical engraving; source and image credit.
  2. Lévi, Transcendental Magic, translated by A. E. Waite, pp. 289–290. Lévi's own account of the forehead pentagram, opposing hands, mercy and justice.
  3. Glorian, "Alchemical Symbolism of the Nativity of Christ", and its teaching on the microcosmic star. Modern Gnostic interpretations of the blessing hand, its shadow and pentagram orientation.
  4. Orthodox Church in America, "Placement of fingers in icons", and "Liturgical Gestures". Christian explanations of IC XC, the Trinity and Christ's two natures.
  5. Homeric Hymn 19, to Pan, translated by Hugh G. Evelyn-White. Ancient literary setting; Perseus Digital Library.
  6. The Met, Marble statue of Pan, first century CE, and Pan or a satyr with winnowing basket, about the second century CE. Classical body and fertility imagery.
  7. Nicolas Poussin, The Triumph of Pan, 1636, National Gallery. Woodland painting reference; the museum notes that its central statue may represent Pan or Priapus.
  8. NET Bible, Job 1:6, translation notes. Hebrew terminology and the adversary’s role in Job.
  9. Ronald Hutton, “The Return of the Horned God,” 2025. Pan’s modern reception and influence on imagery of the Devil.
  10. Julian Strube, "The 'Baphomet' of Eliphas Lévi" (2016), early article PDF. Lévi’s Baphomet, its context and later adaptations.
  11. Joseph P. Laycock, Satanism, Cambridge University Press. Accusation, literature and modern religious identities.
  12. Church of Satan and The Satanic Temple. Primary sources for each organization’s self-description.

The two Pan paintings are newly generated interpretations. The Baphomet image is Lévi's historical engraving, reproduced from Wikimedia Commons. The Christ blessing image is the devotional reference supplied from Christianity Stack Exchange. “The primal pulse of nature” is this gallery’s curatorial language.

Epilogue · The Monad, the Living World and the Star

The One within the many

The gallery began with a divided line. It ends with a question: how can a whole unfold into many forms while remaining one?

The golden cut gives that question a precise geometrical image. Whole and part share one proportion; repeating the construction carries it through smaller rectangles, triangles and stars. The pentagram gathers this relation into fivefold order, each crossing opening the way to another scale. Its continued fraction and nested radical carry the theme into algebra: each expression contains another copy of itself.

φ² = φ + 1   ·   1/φ = φ − 1 One relation, returning through its parts

Pan gives the living world a mythic body: human and animal, musical, fertile and untamed. The later histories of the adversary and the horned image show how that body could acquire other meanings. Lévi's Baphomet gathers opposites into a single figure, its upward and downward hands balancing mercy and justice beneath an upright star.

The turned pentagram and the blessing hand bring another dimension into view: orientation and gesture organize a reading. Spirit and elements, human and animal, above and below become ways to imagine a person's place within the cosmos. The Christian blessing and the modern Gnostic interpretation add their own voices to that history.

For this gallery, the pentagram is the culminating emblem of a pagan and traditional cosmological vision: nature alive, humanity within the cosmos, and multiplicity proceeding from the monad, the principle of unity understood here as divine source. Plato imagined a living cosmos bound by proportion; Plotinus described the many proceeding from the One while their source remains whole.

In that philosophical reading, the golden section becomes the perfect division of unity: a visual language for the monad, or God, expressed through an ordered whole and its related parts. The star returns the eye from the many to the One, and from abstract number to the living world.

Philosophical anchors: Plato, Timaeus, 30b–32c · Plotinus, Enneads, V.2.1–2 · Agrippa, Occult Philosophy, II.27.