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  2. Packing problems - Wikipedia

    en.wikipedia.org/wiki/Packing_problems

    The study of polyomino tilings largely concerns two classes of problems: to tile a rectangle with congruent tiles, and to pack one of each n-omino into a rectangle. A classic puzzle of the second kind is to arrange all twelve pentominoes into rectangles sized 3×20, 4×15, 5×12 or 6×10.

  3. Golden ratio - Wikipedia

    en.wikipedia.org/wiki/Golden_ratio

    A golden rectangle with long side a + b and short side a can be divided into two pieces: a similar golden rectangle (shaded red, right) with long side a and short side b and a square (shaded blue, left) with sides of length a. This illustrates the relationship ⁠ a + b / a ⁠ = ⁠ a / b ⁠ = φ.

  4. Circle packing - Wikipedia

    en.wikipedia.org/wiki/Circle_packing

    The most efficient way to pack different-sized circles together is not obvious. In geometry, circle packing is the study of the arrangement of circles (of equal or varying sizes) on a given surface such that no overlapping occurs and so that no circle can be enlarged without creating an overlap. The associated packing density, η, of an ...

  5. Aspect ratio - Wikipedia

    en.wikipedia.org/wiki/Aspect_ratio

    The aspect ratio of a geometric shape is the ratio of its sizes in different dimensions. For example, the aspect ratio of a rectangle is the ratio of its longer side to its shorter side—the ratio of width to height, [1][2] when the rectangle is oriented as a "landscape". The aspect ratio is most often expressed as two integer numbers ...

  6. Egyptian geometry - Wikipedia

    en.wikipedia.org/wiki/Egyptian_geometry

    The Lahun Papyrus Problem 1 in LV.4 is given as: An area of 40 "mH" by 3 "mH" shall be divided in 10 areas, each of which shall have a width that is 1/2 1/4 of their length. [12] A translation of the problem and its solution as it appears on the fragment is given on the website maintained by University College London.

  7. Rhind Mathematical Papyrus - Wikipedia

    en.wikipedia.org/wiki/Rhind_Mathematical_Papyrus

    Happily the context of 51 and 52, together with the base, mid-line, and smaller triangle area (which are given as 4 + 1/2, 2 + 1/4 and 7 + 1/2 + 1/4 + 1/8, respectively) make it possible to interpret the problem and its solution as has been done here. The given paraphrase therefore represents a consistent best guess as to the problem's intent ...

  8. Dynamic rectangle - Wikipedia

    en.wikipedia.org/wiki/Dynamic_rectangle

    A root rectangle is a rectangle in which the ratio of the longer side to the shorter is the square root of an integer, such as √ 2, √ 3, etc. [2] The root-2 rectangle (ACDK in Fig. 10) is constructed by extending two opposite sides of a square to the length of the square's diagonal. The root-3 rectangle is constructed by extending the two ...

  9. Sphere packing - Wikipedia

    en.wikipedia.org/wiki/Sphere_packing

    In all of these arrangements each sphere touches 12 neighboring spheres, [2] and the average density is π 3 2 ≃ 0.74048. {\displaystyle {\frac {\pi }{3{\sqrt {2}}}}\simeq 0.74048.} In 1611, Johannes Kepler conjectured that this is the maximum possible density amongst both regular and irregular arrangements—this became known as the Kepler ...

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