Sequences and Architecture
Timothy Moulding
Introduction
Looking at certain buildings, donβt you think that there is something particular about some buildings or that some look either staggeringly proportionate or pleasing to your eyes? Well, there actually might be something behind these buildings. That happened to me when I visited the Parthenon in Athens, Greece. The staggering beauty of the Parthenon captured me, but the most interesting part was finding out the mathematics used in building it. With more research, I find out that there are also other buildings that have similar qualities- making them quite appealing as well. In this article, I will discuss what sequences in mathematics are, and explain how sequences have influenced architects to design real buildings out in the world.
What are sequences?
Sequences are lists of numbers that are in order. They follow a specific set of rules and they can be finite and infinite. There are many types of sequences, such as arithmetic sequences, geometric sequences, Fibonacci sequences, etc. In this mathematical investigation, I will explore Fibonacci sequences, the golden ratio, terms in powers of two, and root proportions 1 .
Fibonacci Sequences and The Golden Ratio
A Fibonacci sequence is a special sequence, which goes like this: 0, 1, 1, 2, 3, 5, 8, 13, 21, 34 ... It is a very special sequence: closely related to the Pascalβs triangle and most of all the golden ratio 1 . Phi, π , is known as the golden ratio and it is equal to 1.6180339887498948482...
The golden ratio is defined mathematically, for > > 0 , as:
+
=
= π
If we consider a rectangle with sides 1 and π , then it creates a new square within the rectangle. The new rectangle, green in Figure 1, is similar to the original blue rectangle.
We can see, from the similar properties of the two rectangles, that:
fig. 1: Golden rectangles
π 1
1 π β 1
=
Which gives: π 2 β π β 1 = 0 . This has the unique solution : π = 1+β5 2 = 1.6180β¦ This value is called the golden ratio. A rectangle with such a property is called a golden rectangle. Dividing the rectangle again, leads to a spiral figure that is logarithmic in nature 3 . This is a called a whirling square which can be seen in nature, for example in the construction mollusk shells 4 .
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