Why do Similar Triangles matter?
We have all heard about similar triangles in secondary school.
“Two triangles are similar if the ratio of their corresponding two sides is equal and the angle formed by the two sides is equal”, says the textbook definition (Story of Mathematics).
This is all and well, but why do similar triangles matter in the matrix that is life? Is it just a theory that we learn to be more “well-rounded and rational?” I have not heard anyone debating about similar triangles at cocktail parties, but how would I know? I wouldn’t consider myself to be part of social circles with mathematicians.
In this blog post, I will highlight the influence of similar triangles by taking us on a journey from Ancient Egypt to the present. In the first section, Thales of Miletus will show us how he measured the height of the Great Pyramid of Giza, or the “Cheops Pyramid” in Greek language. In the subsequent sections, I will present two applications of similar triangles which underline the real contributions that geometry has on our lives.
Thales’ Measurement of the Great Pyramid of Giza
Thales of Miletus is known as an all-round talent from Greece who went to Egypt and discovered several geometric properties by following his natural curiosity. At least this is what ancient sources tell us (MathsHistory). Thales figured out how to calculate the height of the pyramids with help from the sun’s rays and returned to Greece to share his exponential growth with the nation.
Hieronymus, who was Aristotle’s student, argues that Thales was able to measure the pyramid’s height naturally by assessing its shadows’ lengths at the point when our own shadow’s length equals our height.
Thales realised that when the length of an object’s shadow equals its height, then this relation must hold for other objects nearby too. Implicitly, Thales was assuming that the angles between the shadows’ lengths and the sun rays were equal for the pyramid and the human. In fact, all the angles seen in the two triangles in the figure must be equal because right angles are captured between the bases and the heights.
Hence, the Greek genius measured the length from the base of the pyramid to the end of its shadow to determine the height of the Great Pyramid of Giza. It seems that Thales’ intuition was playing a larger role than any concrete geometrical knowledge.
Using Thales’ Approach to Measure the Height of Cendana Tower A in terms of my Height
As a disclaimer, COVID-19 restrictions have not allowed me to physically measure the height of the tower. However, I have conceptualised my measurement strategy so that I can execute it once the social distancing regulations in Singapore have eased.
Preparation and Setting
The perfect environment for the operation is a sunny day when we can clearly see the shadows of buildings and objects. We will also need a stretch of yarn that has a length of 1.80 metres, which is known as my height. As Thales has taught us, we need to stand in a position where our own shadow’s length is equivalent to our height. This will only be possible during a certain time of the day, which from empirical observation is between 11AM and 12PM to have the tower’s shadow projected on the Cendana Green, as shown in my diagram below.
Actual Measurement
When the person’s height is the same length as its shadow, then by Thales’ discovery, we know that the length of the tower’s shadow equals the tower’s height.
Now, we line up the 1.80-meter-long pieces of yarn in a row (as indicated on the diagram) and count how many bits of yarn are necessary to form the length of the tower’s shadow.
That’s it! Once we have the number of yarns required, we multiply the amount by 1.80 metres to obtain the height of Cendana Tower A.
Evaluation of Measurement Method
On the bright side, we don’t need to measure any angles which could create potential measurement errors. Determining angles in the field is hard.
Instead of having to climb stairs and estimating the height of differently-designed floors, we just need to compute the length of the tower’s shadow at a particular time of the day. Nature has given us the tools to conduct complex calculations in a simple way.
However, using similar triangles is not without its flaws. There might be measurement errors from using multiple yarns in a line to determine the length of the tower’s shadow. Yarns can extend and contract and controlling for this factor on an uneven surface, i.e. grass, is difficult.
Nevertheless, we can harness the power of geometry to generate accurate approximations of heights. I will update this post as soon as I’ve had the chance to physically measure the tower’s height to prove that I haven’t tricked you.
Similar Triangles in Art
Have you ever tried to draw your apartment as realistically as possible? No? Geometry can help us as it is prevalent in numerous areas of our lives, including art.
Perspective drawing, which came to birth during the early Renaissance period in Italy through artists like Brunelleschi, allows us to illustrate our surroundings in a realistic way. In fact, our brains are naturally accustomed to process images with two points at an infinite distance and all the edges meeting at one of the two points, when we stand at a corner of a room.
In the figure we see an illustration of my apartment at Yale-NUS College with the bathroom in the left half and the corridor on the right. What is fascinating is that the doors’ heights and widths are all in proportion to one another. Thus, the artist can create an illusion that doors further from the eye appear to be smaller, although they obviously all have the same size once we stand directly in front of the doors.
The doors’ sills and heads are connected by a line respectively and the two lines set up two sides of the triangles. The third side is always the height of the door. Hence, the triangles formed by the “smallest door”, the “medium door” and the “largest door” all share the same angle. The best thing is that we do not need to measure the angle to determine the right proportions. This is the magic of similar triangles.
Now you know. The next time you walk in a long hallway or look up to the top of a skyscraper that you admire, remember that similar triangles are at work.