Using a Beautiful Geometric Construction to Review and Practice Angles, Proportionality, and Percents.

You are allowed to do math because it’s beautiful.

This looks similar to our progression of roots, but it’s not. This is a logarithmic spiral of proportional squares. When exhibiting growth through sequences, there are two special types mathematically – arithmetic and geometric. 

Sequential growth is shown with a given distance/number between each series. Think a series of odd numbers 1, 3, 5, 7 … Each time we are adding 2 to the previous number. 

Geometric growth is when the growth is multiplied by a given number. Think about the book A Grain of Rice where there is consistent doubling. 2, 4, 8, 16, 32….

Because the same number is multiplied over and over, there is an exponent involved which makes it a logarithmic sequence as well. 

Just a reminder, logarithms are when we know the base and the final product, but we don’t know the exponent. It is the reverse relationship of an exponent, similar to addition and subtraction  or multiplication and division being inverse operations. Powers – Logarithms 

We have been working with logarithms lately with a review of the pH scale within our chemistry block. I was looking for logarithmic spirals when I came upon this construction of squares. I could tell it was a logarithmic spiral, but it took me a hot second to figure out how the construction was done. 

A logarithmic spiral is also an equal-angular spiral, meaning it stays the same angle from the line at equal angular intervals around the circle. Like all of mathematics, you can explore this from simple to more complex levels. There is definitely trigonometry involved in these spirals, but as we are in eighth grade, and I don’t remember the trig right off, so we are going to explore this through division of a circle, angles, proportions, and percentages. 

The original drawing that I saw was a computer simulation with the circle divided into 36. Division of a circle into 36 is a pretty large number, and also not an easy one to do on paper, in my humble opinion. 

With this construction, I want to practice proportionality through percent decreases, division of a circle, construction of squares, and exploration of angle. These are all concepts that we have gone over before, so this construction will serve as a review and practice.

I went with 24, as division of a circle by six is by far the easiest, and I halved each angle, doubling the division, first 12, and then we arrived at 24. I’ve erased the construction lines so you no longer see the 20 circle divided into 24 pieces. 

I then started with the 10 by 10 square for our first square. From here, each sequential square will be drawn at the next division of the circle line as the base of the next square going in a counterclockwise direction. Each sequential square’s sides decrease by 10%. 

One could continue, but I stopped after each division was filled with a base of a square. 

From here we can elaborate. What are the features that could be changed? The number of divisions which shifts the angles in the circle. The percent decrease or proportion of one square to the next. The polygon that we are drawing. I chose a square, but we could do a series of circles, triangles, pentagons, hexagons. How would these changes alter our drawing? 

This kind of reflection increase our critical thought and our learning and retention. It’s also, dare I say, beautiful!

You’ll notice a series of lines that form a curve at the bottom of the page. After I drew the construction, I copied all the lines in the order of the squares that I constructed from right to left. We can then see what the 10% increase/decrease curve looks like over the construction. 

Exercises like these review concepts, require practice, interleave different mathematical concepts all the while creating something beautiful. You can and probably should explore this through interest as well, but this construction is so much more beautiful. 

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