Ὁ Πενταμετρικὸς Κύκλος – The Pentametric Cycle

Ὁ Πενταμετρικὸς Κύκλος

The Pentametric Cycle

𝔓(n)♪(n)
𝔓(n) = n² + 4 C(m)5, m = 4n² mod 10, C(m) = 26m − 5m²24
𝔓(Lk) = Fk²

The formula as found. C(m) equals the Legendre symbol n5 (0 or ±1, not a fraction), so 𝔓(n) = (n² + 4·n5) / 5. ♪(n) is the interval of 𝔓(n) mod 5.

A formula found by experiment on the squares of the Fibonacci spiral. At every Lucas number 1, 3, 4, 7, 11, 18, … it returns the square of the matching Fibonacci number, and nowhere else is it a perfect square. Reduced modulo 5, it becomes a palindrome of the four Pythagorean consonances: unison, octave, fifth and fourth.

◆ Understand

Interactive explanations of the concepts and results in the papers.

Geometry

The spiral and Holden's lines

Fibonacci squares, the point P, and a square through P whose area is Fk2 ± 4/5.

 Hidden grid

Thirds and tenths

Holden's lines cut every side at ⅓ and ⅔ and meet on tenths: a tilted grid of origins units (√0.1) hidden in the tiling. Measure any distance exactly.

 Correction

The ±0.8

How n²/5 is moved by ±0.8 to a whole number, why the circles move by ±2π/5, and the last-digit rule.

 Why five

The pentagon clock

Step around a regular pentagon: the vertex decides the sign of the correction, and the pentagon's mirror makes the palindrome.

 Pythagoras

Monochord and tetractys

Pluck a string at ½, ⅔ and ¾ and see the four consonances as neighbouring rows of the tetractys.

 Music

The playable palindrome

The 51-step cycle as a wheel and a piano roll that you can hear in just intonation.

 Arithmetic

Formula explorer

Follow any number through the seven steps to its area and interval, and search for perfect squares.

 Exact relations

Identities gallery

Twelve identities of the construction, from L² − 5F² = ±4 to 9 + 16 = 25, each checked live for any k or n.

 Ten routes

Ten forms of one formula

Digits, floors, a quintic, cosines, a golden rotation and pentagon angles, all giving the same areas.

 Animation · 2D and 3D

The Pentametric spiral

Squares of area 𝔓(n) turning 72° per step around a pentagon. The Lucas squares swing into the golden rectangle, the Fibonacci spiral grows inside it with sound, and in 3D everything grows in time.

 Note · 5 : 4

The Epitetartic Rhombus

A rhombus with diagonals 5 : 4, found where Holden's line meets an arc of the spiral, and held by every square: its construction with a ruler, and why halving never finds its corner while threes and fives do.

 Note · distance

Distance to the spiral

Move along the Fibonacci spiral and watch its distance from the Origin in origins units: whole numbers exactly on Holden's lines (2, 4, 6, 10, 16, …), Pythagoras at the joins, growth by φ.

 Note · origins

The origin of origins

Of all the circles about the centre of the unit square, only one has a tangent through a corner where it crosses a curved diagonal: area π/10. Its eight crossings are the origins of eight Fibonacci spirals, and Holden's lines are its tangent and radius there.

 Note · cells

Counting the areas

From an empty square and a ruler: the square of area 1/5, the circle of origins, Holden's lines and a grid of cells of 1/10 nobody drew. Then 𝔓(n) is a count of cells: the blue square of n, plus or minus the blue square of 2.

 Note · landing

Where the origin lands

Step Holden's crossing out, H, 2H, 3H, … and it always lands exactly on an origin of a unit square, turned 0, ¾, ¼, ½ of a turn, or on a corner every fifth step: the breath as a place, and a quarter turn per Fibonacci square.

 Note · hidden origin

The hidden origin

Run the Fibonacci spiral backwards with squares of half the size: it leads in smoothly from H′ = (7/10, −1/10), the mirror of Holden's crossing and an origin of the next square. Other ratios, the lines of slopes 7 and −1/7, and the golden pole.

✦ Exploratory

Ideas that go beyond the papers. They are shown as explorations and observations, not as published results.

Exploratory

Other primes

Why 5 is the only prime for the formula, and how the palindrome and the triple 5² + 12² = 13² appear for other primes.

 Music

The 13-cycle as music

With 13 instead of 5 the palindrome sounds every superparticular ratio from 2:1 to 12:11. Listen to it.

 Residues

Higher powers of 5

The cycle modulo 25, 125, 625: periods, mirrors and the changing pattern of zeros.

 Other sequences

Pell numbers

Pell numbers obey the same kind of identity. Can a remainder rule pick the sign again? A search says no so far.

 Structure

The 24 transitions

Which interval follows which: 17 pairs out of 25, as a table and a network you can hear.

 Curiosity

The ratio √10/3

A hypotenuse of the construction that lies within 1 cent of the Pythagorean semitone. Listen and compare.

 Exploratory

The angle of 0.8

The ±0.8 correction as an angle, arctan(4/5): one triangle read three ways, and the family of angles whose primes give a square number of silences.

 Exploratory

When silence becomes 5

Lluis reads the groups of five as 1–5: the silences become major thirds 5:4, and the steps between values play the harmonics 1–5, like a natural trumpet.

 Exploratory · sound

The chord wheel

Lay the twelve notes on a quarter of the palindrome: every chord becomes a polygon. Measure its angles, hear it, and see which values lie under its notes.

 Exploratory · sound

The palindrome on Euler's lattice

The notes of just intonation as a lattice of fifths and thirds: watch the palindrome walk on it, every leap paired with its mirror, and the comma open between just and 12-tone roots.

 Exploratory · tool

The mirror fretboard

In all-fourths tuning the guitar is a mirror: reflect the board through a root and every interval lands on its inversion. Play the palindrome on it.

 Exploratory · 3D

The palindrome on the Tonnetz torus

Fold the flat Tonnetz into a tube and a torus, turn the mirror, walk the cycle step by step, and meet the six notes of the Pentametric areas.

✧ Exotic explorations

Playful explorations of old symbolic traditions around the cycle: astrology, magic squares and more. They are shown for fun and curiosity, not as claims.

Exotic · sky

The sky wheel

The planets on the zodiac for any date and place, their aspects, the palindrome under the signs, and the chord the sky makes.

⤓ Use it

Tools that make something you can take away: music as MIDI files, and images to print or share.

Tool

Spiral Workbench

The eight Fibonacci spirals with their squares, Holden's lines and grids, and tools to draw and measure. Turn the spirals about their origins and watch where they cross, with trails and a graph of any measurement against the angle.

 Make music

Composer and MIDI export

Turn any list of numbers into a piece of consonances, listen, and download it as a MIDI file.

 Pattern

The residue tapestry

Weave the sequence into rows of any length and download the pattern as a high-resolution image.

 Personal

Name and date signature

Every name and date has its own short piece in the cycle. Listen, then download it as MIDI or as a card to share.

 Tuning

Tuning file maker

The cycle's scales, from the tetractys to the Pythagorean chromatic, as Scala (.scl) files for synthesizers.

 Drums

Rhythm generator

The cycle as a drum pattern that drifts against the bar. Edit it, play it, download it as MIDI.

 Music · MIDI

Chord progressions

The cycle writes the harmony: only I, IV and V, coming home every period. Smooth voicings, a bass line, flamenco rumba and compás, MIDI download.

 Print

Poster maker

The wheel, the spiral or the tapestry in six palettes, as SVG or PNG for prints, wallpapers and covers.

◇ Library

Read

The papers

The preprint, the three notes and the palindrome data sheet, to read here or download.

 See

The images

The GeoGebra drawings the findings came from: the Origin, Holden's lines, the ±0.8, the unit square and the rhombus.

 Explore

The GeoGebra files

The constructions themselves, live in the page: drag the points, zoom in, and download any of the 61 files.

Paper, data and credits

Text and figures CC BY 4.0; code MIT. These pages were built with the help of an AI assistant (Claude, Anthropic); the mathematics is the author's and every number shown is computed live in your browser.