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Journey 10 of 10 · 9 steps

The Builders’ Measures

Rope, compass and the path

The proportions that surveyors, masons and pavement-layers could make with a knotted rope and a compass: the right angle, the golden cut, the octagon, and a path to walk.

Play it in SacredForm

In the app each step moves: figures draw themselves, rise from the page, turn and slice, and the narration can be read aloud. Here they are held still, one picture each.

  1. Twelve knots: Egyptian Triangle

    Step 1 · Egyptian Triangle

    Twelve knots

    Stretch a loop of twelve equal spans round three pegs as three, four and five: the corner between the three and the four is square. No protractor needed.

  2. Why it works: Egyptian Triangle

    Step 2 · Egyptian Triangle

    Why it works

    The squares on its sides hold nine, sixteen and twenty-five cells, and nine and sixteen make twenty-five: Pythagoras, in whole numbers.

  3. The golden right angle: Kepler Triangle

    Step 3 · Kepler Triangle

    The golden right angle

    Now let the sides grow by the same factor each time: 1, √φ, φ. The squares on them are 1, φ and φ², and since φ² = φ + 1 the triangle is right-angled.

  4. A pyramid: The Kepler Pyramid

    Step 4 · Kepler Triangle

    A pyramid

    Put two back to back and turn them about the height: a square pyramid whose faces slope at 51.83°, within a hair of the Great Pyramid’s.

  5. The golden triangle: Golden Triangle

    Step 5 · Golden Triangle

    The golden triangle

    The point of a pentagram, 36° at the top. Cut it from a base corner and it splits into a smaller copy and a golden gnomon. Cut again and the copies whirl inward.

  6. The sacred cut: The Sacred Cut

    Step 6 · The Sacred Cut

    The sacred cut

    Open the compass from a corner of a square to its centre and swing it to the sides. Four arcs cut the square into nine and mark a perfect octagon, with no measuring at all.

  7. A face of a solid: Truncated Cube

    Step 7 · The Sacred Cut

    A face of a solid

    That octagon is a face of the truncated cube: lay the cut on every face of a cube and slice the corners away.

  8. The magic square: Lo Shu Square

    Step 8 · Lo Shu Square

    The magic square

    Nine numbers so that every line of three sums to fifteen. Join them in order and a sigil appears, the one a turtle is said to have carried out of the Luo River.

  9. A path to walk: Chartres Labyrinth

    Step 9 · Chartres Labyrinth

    A path to walk

    Set into the floor of Chartres Cathedral eight centuries ago: eleven rings and one path. It reaches the innermost ring at once, swings out to the edge, and only then comes to the centre. Watch it walk. Draw with a compass in Construct