---
jupyter:
description: 'Geometry and design examples with planex2d in Python: tilings, fractals and packings, classic
theorems from Archimedes to Pick, the moving sofa and Prince Rupert''s cube, curves from the Bézier circle to
the squircle, and the section properties of steel profiles.'
---
# Geometry and design examples
- [Penrose tiling](penrose-tiling.md)
- [Rose window](rose-window.md)
- [Path planning](path-planning.md)
- [The moving sofa](moving-sofa.md)
- [Prince Rupert's cube](prince-ruperts-cube.md)
- [The coastline paradox](coastline-paradox.md)
- [The pole of inaccessibility](pole-of-inaccessibility.md)
- [Section properties of steel profiles](section-properties.md)
- [The spirograph](spirograph.md)
- [The Hausdorff distance](hausdorff-distance.md)
- [The isoperimetric inequality](isoperimetric-inequality.md)
- [The lune of Hippocrates](lune-of-hippocrates.md)
- [Circle packing in a square](circle-packing.md)
- [The problem of Apollonius](problem-of-apollonius.md)
- [Drawing a logo](drawing-a-logo.md)
- [Why bees build hexagons](honeycomb.md)
- [Vaults and Steinmetz solids](vaults.md)
- [Gabriel's horn](gabriels-horn.md)
- [Buffon's needle](buffons-needle.md)
- [Pick's theorem](picks-theorem.md)
- [The Steiner tree problem](steiner-tree.md)
- [Sunflower seeds and the golden angle](sunflower-golden-angle.md)
- [Drawing with epicycles](drawing-with-epicycles.md)
- [Kepler's second law](keplers-second-law.md)
- [The Bézier circle](bezier-circle.md)
- [The Sierpiński triangle](sierpinski-triangle.md)
- [Square packing](square-packing.md)
- [The Koch snowflake](koch-snowflake.md)
- [Lissajous curves](lissajous-curves.md)
- [The golden spiral and the Fibonacci spiral](golden-spiral.md)
- [The squircle and the superellipse](squircle.md)
- [Squaring the square](squaring-the-square.md)
- [The pizza theorem](pizza-theorem.md)
- [The Seven Bridges of Königsberg](seven-bridges.md)
These examples are problems of plane geometry with a known answer, from Archimedes and Hippocrates to the moving sofa
and the packing of circles and squares. planex2d keeps lines, arcs and circles exact, so a result can be checked to
many digits. The lune of Hippocrates has exactly the area of its triangle, Prince Rupert's cube lets a cube of side
$3\sqrt{2}/4$ pass, and the Koch snowflake encloses 8/5 of its triangle. Others are design tasks, as a rose window
from circles that touch, the planex2d logo from its own SVG and the section properties of steel profiles against the
tables.
In this section
Penrose tilingGolden triangles deflated, with the area kept exactly and the thick-to-thin ratio tending to φ.
Rose windowGothic tracery from circles that touch circles, bars as offsets.
Path planningObstacles grown by offsets, and the shortest path through a visibility graph.
The moving sofaThe largest sofa around a corner as one intersection, Hammersley's exactly and Gerver's within 0.00012.
Prince Rupert's cubeA cube through a hole in a cube of its size, the largest one 3√2/4 to twelve digits.
The coastline paradoxRichardson's dividers on the Koch snowflake and on the coast of Great Britain.
The pole of inaccessibilityThe point of Great Britain farthest from the sea, as the largest circle inside its coast.
Section properties of steel profilesSecond moments of area, an IPE 200 and an unequal angle against the steel tables, Mohr's circle.
The spirographWhen a pattern closes and how many petals it has, its length as an elliptic integral, the deltoid and the astroid.
The Hausdorff distanceHow far a polygon lies from its circle, Archimedes' 96-gons and the chord tolerance of a tessellation.
The isoperimetric inequalityWhy the circle encloses the most: quotients of shapes against their exact values, and Dido's problem.
The lune of HippocratesA crescent of exactly a triangle's area as one Boolean difference, and the lunes of Alhazen.
Circle packing in a squareEqual circles spread by SciPy, checked as exact discs against the proven best packings.
The problem of ApolloniusCircles tangent to three given circles from a constraint sketch: Descartes' two and all eight.
Drawing a logoThe planex2d mark from its own SVG: Bézier curves as exact B-splines, a Reuleaux triangle of constant width.
Why bees build hexagonsCells of equal area glued: the hexagon needs the least wall, half its perimeter per cell.
Vaults and Steinmetz solidsGroin and cloister vaults from two barrels, crossed cylinders and the sphere inside.
Gabriel's hornA horn of finite volume and endless surface, from one exact rational B-spline.
Buffon's needleNeedles dropped on lines estimate π, and a bent noodle crosses 2L/(πd) lines whatever its shape.
Pick's theoremThe area of a polygon on a grid from the points inside it and on its boundary, with and without holes.
The Steiner tree problemThe shortest network joining points, with 120° junctions placed by the solver, checked to twelve digits.
Sunflower seeds and the golden angleVogel's model: at the golden angle the seeds keep their distance, a tenth of a degree off they crowd.
Drawing with epicyclesThe letter π drawn by a chain of rotating circles, its area from the Fourier coefficients.
Kepler's second lawEqual areas in equal times: sectors of an orbit from Kepler's equation, cut from the exact ellipse.
The Bézier circleFour cubic quarters off the circle by at most 0.027 %, a better handle, and the exact rational quarter.
The Sierpiński triangleMiddle triangles cut out until the area is gone and the boundary endless, and the carpet full of holes.
Square packing5, 10 and 17 unit squares in the smallest known squares, checked free of overlaps.
The Koch snowflakeAn endless boundary around 8/5 of a triangle, inside a circle, checked step by step.
Lissajous curvesThe oscilloscope curves cut at their own crossings, 2ab − a − b of them, and the frequency ratio from the touches.
The golden spiral and the Fibonacci spiralFibonacci squares, their quarter circles and the true golden spiral a hundredth of the radius away.
The squircle and the superellipseSuperellipse areas from the Gamma function, and why a squircle looks smoother than a rounded square: its curvature.
Squaring the squareDuijvestijn's 21 different squares from their Bouwkamp code, and the electrical network whose currents are their sides.
The pizza theoremEight slices cut through any point share pizza and crust equally; with other cuts, the center decides who gets more.
The Seven Bridges of KönigsbergThe graph read from a map of the river and its bridges: Euler's criterion for the seven, and a walk over the five of today.
See [Examples](index.md) for the other collections.
