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In geometry, a golden rectangle is a rectangle with side lengths in golden ratio +:, or :, with approximately equal to 1.618 or 89/55. Golden rectangles exhibit a special form of self-similarity : if a square is added to the long side, or removed from the short side, the result is a golden rectangle as well.
where φ = 1 + √ 5 / 2 is the golden ratio. Therefore, the circumradius of this rhombicosidodecahedron is the common distance of these points from the origin, namely √ φ 6 +2 = √ 8φ+7 for edge length 2. For unit edge length, R must be halved, giving R = √ 8φ+7 / 2 = √ 11+4 √ 5 / 2 ≈ 2.233.
Hambidge distinguishes these from rectangles with rational proportions, which he terms static rectangles. [3] According to him, root-2, 3, 4 and 5 rectangles are often found in Gothic and Classical Greek and Roman art, objects and architecture, while rectangles with aspect ratios greater than root-5 are seldom found in human designs. [4]
Stacking golden rectangles produces golden rectangles anew, and removing or adding squares from golden rectangles leaves rectangles still proportioned in ratio. They can be generated by golden spirals , through successive Fibonacci and Lucas number-sized squares and quarter circles.
Pacioli points out that golden rectangles can be inscribed by an icosahedron, [6] and in the fifth chapter, gives five reasons why the golden ratio should be referred to as the "Divine Proportion": [7] Its value represents divine simplicity. Its definition invokes three lengths, symbolizing the Holy Trinity.
Many works of art are claimed to have been designed using the golden ratio. However, many of these claims are disputed, or refuted by measurement. [1] The golden ratio, an irrational number, is approximately 1.618; it is often denoted by the Greek letter φ .
Apollonius of Perga discovered the curious result that the ratio of volumes of these two shapes is the same as the ratio of their surface areas. [12] Both volumes have formulas involving the golden ratio but are taken to different powers. [1] Golden rectangle may also related to both regular icosahedron and regular dodecahedron. The regular ...
Rabatment of the rectangle is a compositional technique used as an aid for the placement of objects or the division of space within a rectangular frame, or as an aid for the study of art. Every rectangle contains two implied squares, each consisting of a short side of the rectangle, an equal length along each longer side, and an imaginary ...