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  2. 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.

  3. 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.

  4. 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 ...

  5. Square packing - Wikipedia

    en.wikipedia.org/wiki/Square_packing

    Square packing in a square is the problem of determining the maximum number of unit squares (squares of side length one) that can be packed inside a larger square of side length . If is an integer, the answer is but the precise – or even asymptotic – amount of unfilled space for an arbitrary non-integer is an open question. [1] The smallest ...

  6. Rectangle - Wikipedia

    en.wikipedia.org/wiki/Rectangle

    Rectangle. In Euclidean plane geometry, a rectangle is a quadrilateral with four right angles. It can also be defined as: an equiangular quadrilateral, since equiangular means that all of its angles are equal (360°/4 = 90°); or a parallelogram containing a right angle. A rectangle with four sides of equal length is a square.

  7. Golden rectangle - Wikipedia

    en.wikipedia.org/wiki/Golden_rectangle

    A golden rectangle with sides ab placed adjacent to a square with sides of length a produces a similar golden rectangle. In geometry, a golden rectangle is a rectangle whose side lengths are in the golden ratio, , which is (the Greek letter phi), where is approximately 1.618. Golden rectangles exhibit a special form of self-similarity: All ...

  8. Plastic ratio - Wikipedia

    en.wikipedia.org/wiki/Plastic_ratio

    The internal, coincident short edge of the medium rectangle and long edge of the small rectangle divides one of the square's other, two edges into two segments that stand to one another in the ratio ρ 4. The fact that a rectangle of aspect ratio ρ 2 can be used for dissections of a square into similar rectangles is equivalent to an algebraic ...

  9. 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 ...

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