Geometry and design examples
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
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Penrose tilingGolden triangles deflated, with the area kept exactly and the thick-to-thin ratio tending to φ.
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Rose windowGothic tracery from circles that touch circles, bars as offsets.
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Path planningObstacles grown by offsets, and the shortest path through a visibility graph.
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The moving sofaThe largest sofa around a corner as one intersection, Hammersley's exactly and Gerver's within 0.00012.
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Section properties of steel profilesSecond moments of area, an IPE 200 and an unequal angle against the steel tables, Mohr's circle.
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The spirographWhen a pattern closes and how many petals it has, its length as an elliptic integral, the deltoid and the astroid.
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The Hausdorff distanceHow far a polygon lies from its circle, Archimedes' 96-gons and the chord tolerance of a tessellation.
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The isoperimetric inequalityWhy the circle encloses the most: quotients of shapes against their exact values, and Dido's problem.
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The lune of HippocratesA crescent of exactly a triangle's area as one Boolean difference, and the lunes of Alhazen.
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Circle packing in a squareEqual circles spread by SciPy, checked as exact discs against the proven best packings.
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The problem of ApolloniusCircles tangent to three given circles from a constraint sketch: Descartes' two and all eight.
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Why bees build hexagonsCells of equal area glued: the hexagon needs the least wall, half its perimeter per cell.
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Buffon's needleNeedles dropped on lines estimate π, and a bent noodle crosses 2L/(πd) lines whatever its shape.
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Pick's theoremThe area of a polygon on a grid from the points inside it and on its boundary, with and without holes.
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The Steiner tree problemThe shortest network joining points, with 120° junctions placed by the solver, checked to twelve digits.
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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.
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Drawing with epicyclesThe letter π drawn by a chain of rotating circles, its area from the Fourier coefficients.
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Kepler's second lawEqual areas in equal times: sectors of an orbit from Kepler's equation, cut from the exact ellipse.
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The Bézier circleFour cubic quarters off the circle by at most 0.027 %, a better handle, and the exact rational quarter.
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The Sierpiński triangleMiddle triangles cut out until the area is gone and the boundary endless, and the carpet full of holes.
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Square packing5, 10 and 17 unit squares in the smallest known squares, checked free of overlaps.
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The Koch snowflakeAn endless boundary around 8/5 of a triangle, inside a circle, checked step by step.
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Lissajous curvesThe oscilloscope curves cut at their own crossings, 2ab − a − b of them, and the frequency ratio from the touches.
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The golden spiral and the Fibonacci spiralFibonacci squares, their quarter circles and the true golden spiral a hundredth of the radius away.
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The squircle and the superellipseSuperellipse areas from the Gamma function, and why a squircle looks smoother than a rounded square: its curvature.
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Squaring the squareDuijvestijn's 21 different squares from their Bouwkamp code, and the electrical network whose currents are their sides.
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The pizza theoremEight slices cut through any point share pizza and crust equally; with other cuts, the center decides who gets more.
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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 for the other collections.