of two are applied to architecture, the is a non-negative number. Each positive power of two is two times the previous 11 . Below I show the factors of two explicitly by writing:
2, 2 · 2, 2 · 2 · 2, 2 · 2 · 2 · 2, 2 · 2 · 2 · 2 · 2, 2 · 2 · 2 · 2 · 2 · 2, …
Using exponents, I can rewrite it as:
2 1 , 2 2 , 2 3 , 2 4 , 2 5 , 2 6 , …
And more concisely as:
{2 | 𝑖 𝑖𝑖𝑣 𝑖}
The Alhambra
The Palace of the Alhambra in Granada, Spain, (as shown in Figure 7) was once the residence of the Moorish rulers built in the 14 th century. The Sultana’s quarters have an interesting infinite sequence, as though it is possible to carry on indefinitely. The Moorish architects designed the rooms in the palace so that there were two marble slabs inlaid in the floor, four walls, an octagon ceiling, 16 windows, 32 arches and so on. Here we have an infinite sequence in the powers of 2: 2, 4, 8, 16, 32 … Design constraints would eventually stop the sequence 12 .
fig.7: The Alhambra
The architects used the terms in powers of two in order to create symmetries and patterns to show the divine beauty representing the divine of God. The intention of the Moorish architects to use the terms of powers of two differed from those architects who built by Fibonacci sequence and golden ratio discussed previously. Rather than showing the physical beauty of the body, the Moorish architects applied the terms in powers of two in serving an abstract intellectual beauty of God that forced the intellects to engage with the world of the divine 13 . This reflects the spiritual nature of mathematics in the Moorish architecture.
Root Proportions
Root proportions, another sequence, is a proportion in which the ratio of the longer side to the shorter is the square root of an integer, such as √2 , √3 , etc. The √2 rectangle is constructed by extending two opposite sides of a square (the square on the left hand side of √2 in fig. 8) to the length of the square’s diagonal. The root-3 rectangle is constructed by extending the two longer sides of a root-2 rectangle to the length of the root-2 rectangle’s diagonal. Each successive root rectangle is produced by extending a root rectangle’s longer sides to equal the length of that rectangle’s diagonal 14 . As a result, a sequence of roots derived from the square is illustrated in Figure 8 below with each red line being the root.
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