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Compass and straightedge constructions

Twenty lessons in the spirit of Euclid

The rules are the oldest in geometry. A straightedge draws a line through two points. A compass draws a circle about one point through another. New points are only ever where lines and circles cross. With nothing else, the Greeks built the regular polygons, the square roots of two and three, and the golden section.

Each lesson below is drawn step by step, from the two circles of the vesica piscis to the hidden cube of Metatron’s Cube. In the app’s Construct mode you can draw them yourself, and after them come 22 challenges with no teaching at all.

  1. 1Vesica PiscisDraw two circles, each centred on the other’s rim.
  2. 2Equilateral TriangleBuild an equilateral triangle on the segment AB.
  3. 3Perpendicular BisectorDraw the line that cuts AB in half at a right angle.
  4. 4The SquareBuild a square on the segment AB.
  5. 5The Rope-StretchersStretch the rope into a right angle at A: a triangle with sides 3, 4 and 5.
  6. 6The HexagonInscribe a regular hexagon in the circle, starting at P.
  7. 7Seed of LifeSurround the circle with six circles of the same size.
  8. 8The DodecagonInscribe a regular twelve-sided polygon, starting at P.
  9. 9Angle BisectorCut the angle at V into two equal angles.
  10. 10The Root of ThreeDraw the vesica and join its two crossings.
  11. 11The Root of TwoSwing the square’s diagonal down onto the line of its base.
  12. 12The Sacred CutSwing an arc from each corner through the centre, and join the cuts into an octagon.
  13. 13The Golden CutDivide AB at G so that AB : AG = AG : GB.
  14. 14Golden RectangleExtend the square into a golden rectangle.
  15. 15The Kepler TriangleBuild a right triangle on AB, square at B, whose longest side is φ.
  16. 16The PentagonInscribe a regular pentagon with a corner at T.
  17. 17The PentagramJoin every second corner to draw the star.
  18. 18Dürer’s HeptagonStep Dürer’s near-heptagon around the circle from P.
  19. 19Flower of LifeGrow the Seed into the Flower: add the next twelve circles.
  20. 20Metatron’s CubeFind the cube and the star tetrahedron hidden in the Fruit of Life.