1) The angle of elevation of the slant height of Egyptian pyramids is commonly 44° or 52°. These angles allegedly provide
an aesthetic quality to the pyramids. Do you agree that these are the most beautiful (aesthetically pleasing) angles?
Each of the four triangular-shaped sides of a pyramid must slope up and towards each other at the same angle so that
they meet at a point at the top. Also, the sides are believed to be aligned true north. For this to happen, the slant height
of the pyramid should form a golden ratio (φ) with the semi-base, as shown in the figure below …
The approximate value of the golden ratio φ is 1.618. Now, 1/ φ = 0.618, and the cosine of this angle is cos α = 51.83 or
52°.
Thus, the angle 52° is correlated to the golden ratio, which according to history and available research results is
considered pleasing. There are hypotheses that golden ratio is observed in many natural situations, including the
proportions of a human body. But such claims have not been conclusively proven to be true. However, aesthetic aspect
apart, the golden ration appears to be architecturally robust, and this could be one of the reasons that the great
pyramids have withstood the test of time and elements, and are standing even today.
The angle 44° that has been used in some pyramids appears to be another choice from a structural point of view. (A
pyramid with a square base and its apex exactly in a vertical line with the center of the base has a limited number of
angles to choose from).
2) Determine whether matrix multiplication is commutative. Prove your answer using an example.
Matrix multiplication is not commutative in general. Here is an example to prove this …
1
Let 𝐴 = [
2
3
−1 5
] 𝑎𝑛𝑑 𝐵 = [
]
6
4 9
First we do AB
Now, we do BA
We see that AB ≠ BA. This means matrix multiplication is not commutative in general.