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  2. Pythagorean theorem - Wikipedia

    en.wikipedia.org/wiki/Pythagorean_theorem

    In mathematics, the Pythagorean theorem or Pythagoras' theorem is a fundamental relation in Euclidean geometry between the three sides of a right triangle.It states that the area of the square whose side is the hypotenuse (the side opposite the right angle) is equal to the sum of the areas of the squares on the other two sides.

  3. Silver ratio - Wikipedia

    en.wikipedia.org/wiki/Silver_ratio

    The parallelogram between the pair of grey triangles on the sides has perpendicular diagonals in ratio ⁠ ⁠, hence is a silver rhombus. If the triangles have legs of length ⁠ 1 {\displaystyle 1} ⁠ then each discrete spiral has length σ σ − 1 = ∑ n = 0 ∞ σ − n . {\displaystyle {\frac {\sigma }{\sigma -1}}=\sum _{n=0}^{\infty ...

  4. Rhombus - Wikipedia

    en.wikipedia.org/wiki/Rhombus

    A rhombus has an inscribed circle, while a rectangle has a circumcircle. A rhombus has an axis of symmetry through each pair of opposite vertex angles, while a rectangle has an axis of symmetry through each pair of opposite sides. The diagonals of a rhombus intersect at equal angles, while the diagonals of a rectangle are equal in length.

  5. Golden rectangle - Wikipedia

    en.wikipedia.org/wiki/Golden_rectangle

    The respective lengths a, b, and c of the sides of these three polygons satisfy the equation a 2 + b 2 = c 2, so line segments with these lengths form a right triangle (by the converse of the Pythagorean theorem). The ratio of the side length of the hexagon to the decagon is the golden ratio, so this triangle forms half of a golden rectangle. [8]

  6. Golden ratio - Wikipedia

    en.wikipedia.org/wiki/Golden_ratio

    The triangle formed by two diagonals and a side of a regular pentagon is called a golden triangle or sublime triangle. It is an acute isosceles triangle with apex angle ⁠ ⁠ and base angles ⁠ ⁠. [45] Its two equal sides are in the golden ratio to its base. [46]

  7. Arrangement of lines - Wikipedia

    en.wikipedia.org/wiki/Arrangement_of_lines

    Roberts's triangle theorem and the Kobon triangle problem concern the minimum and maximum number of triangular cells in a Euclidean arrangement, respectively. Algorithms in computational geometry are known for constructing the features of an arrangement in time proportional to the number of features, and space linear in the number of lines.

  8. Pythagorean prime - Wikipedia

    en.wikipedia.org/wiki/Pythagorean_prime

    The formulas show how to transform any right triangle with integer legs into another right triangle with integer legs whose hypotenuse is the square of the first triangle's hypotenuse. A Pythagorean prime is a prime number of the form 4 n + 1 {\displaystyle 4n+1} .

  9. Rhombille tiling - Wikipedia

    en.wikipedia.org/wiki/Rhombille_tiling

    When more than one type of rhombus is allowed, additional tilings are possible, including some that are topologically equivalent to the rhombille tiling but with lower symmetry. Tilings combinatorially equivalent to the rhombille tiling can also be realized by parallelograms, and interpreted as axonometric projections of three dimensional cubic ...