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A magic triangle or perimeter magic triangle[1] is an arrangement of the integers from 1 to n on the sides of a triangle with the same number of integers on each side, called the order of the triangle, so that the sum of integers on each side is a constant, the magic sum of the triangle. [1][2][3][4] Unlike magic squares, there are different ...
The primary square is obtained by rotating the root square counter-clockwise by 90 degrees, and replacing the numbers. The resulting square is an associative magic square, in which every pair of numbers symmetrically opposite to the center sum up to the same value, 26. For e.g., 16+10, 3+23, 6+20, etc.
Magic circle (mathematics) Yang Hui's magic concentric circles – numbers on each circle and diameter (ignoring the middle 9) sum to 138. Magic circles were invented by the Song dynasty (960–1279) Chinese mathematician Yang Hui (c. 1238–1298). It is the arrangement of natural numbers on circles where the sum of the numbers on each circle ...
Pythagorean theorem. The sum of the areas of the two squares on the legs (a and b) equals the area of the square on the hypotenuse (c). In mathematics, the Pythagorean theorem or Pythagoras' theorem is a fundamental relation in Euclidean geometry between the three sides of a right triangle.
The apparent triangles formed from the figures are 13 units wide and 5 units tall, so it appears that the area should be S = 13×5 / 2 = 32.5 units. However, the blue triangle has a ratio of 5:2 (=2.5), while the red triangle has the ratio 8:3 (≈2.667), so the apparent combined hypotenuse in each figure is actually bent.
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]
The earliest extant Chinese illustration of 'Pascal's triangle' is from Yang's book Xiangjie Jiuzhang Suanfa (詳解九章算法) [1] of 1261 AD, in which Yang acknowledged that his method of finding square roots and cubic roots using "Yang Hui's Triangle" was invented by mathematician Jia Xian [2] who expounded it around 1100 AD, about 500 years before Pascal.
The circle packing theorem (also known as the Koebe–Andreev–Thurston theorem) describes the possible tangency relations between circles in the plane whose interiors are disjoint. A circle packing is a connected collection of circles (in general, on any Riemann surface) whose interiors are disjoint. The intersection graph of a circle packing ...