Mark’s Fractal Geometry Problem

We featured a math quizzer from De La Salle University in our previous post. Now let’s go to Mapua Institute of Technology to feature one of their quizzers named Mark Elis Espiridion.

I never had a huge fascination with Mathematics when I was younger. I tend to like Science more than any subject. My “love story” with Math began when I was in Grade 5 when my Math teacher called my attention for answering our Math exercises very quickly (I tend to do it while she was discussing the topic halfway). It was a more of an ellis pixon-off relationship from then on. We had fights, and we made up. There are times when I didn’t want to do Math, but there were times when Math was just the only thing I could think about. But the peak of our relationship is when I entered college, I joined a specific math contest inside our campus. It was the second time I joined a math contest, specifically, it was about statistics and probability. During the encounter, I was really having a hard time answering the questions since I depended on my stocked knowledge. But in the end, it’s like a match made in heaven. I won my very first academic contest inside the institute. It felt so good like you’re floating on cloud nine. That was the beginning of my “career” as a quizzer.” Said Mark.

“I am currently taking up Bachelor of Science in Civil Engineering in the Mapua Institute of Technology. Besides academics, I also spent time training in sports. I was also a swimmer and had competed in different swim meets with our swim club. I even attended a try-out to the Philippine Team on Fin Swimming but I didn’t pursue it because of a possible home schooling requirement. I think I got my competitive spirit from this sport.” He added.

As a quizzer, here are some of his achievements.

In-Campus Contests:

Mapua Statistics Quiz for Freshmen (2010) – Champion

Mapua Math Wizards (2011) – 4th Runner Up

PICE-MIT Quiz Show (2012) – 1st Runner Up

Mapua Statistics Quiz For Higher Years (2012) – Champion

ACIMIT-SC Mechanics Quiz (2012) – 2nd Runner Up

Mapua Math Wizards (2012) – Champion

PSM Physics Quiz Bee (2013) – Finalist

PICE-MIT Quiz Show (2013) – Champion

ACIMIT-SC Mechanics Quiz (2013) – 2nd Runner Up

Out-Campus Contest:

NSO Statistics Quiz (2010) – Finalist

UPLB-CEO Executives Challenge 8 (2013) – Finalist

MSP Annual Search for Math Wizard (2013) – Semi-finalist

PaCEkatan CE Quiz (2013) – Finalist

UPLB-CEO Executives Challenge 9 (2014) – 1st Runner Up

MSP Annual Search for Math Wizard (2014) – Semi-finalist

And now, here is a very elegant problem on geometry created by Mark himself. He also contributed similar problem to brilliant.org.

elis4

The Problem:

Find the total area of all the circles that is drawn endlessly following a certain pattern if the radius of the largest circle is one.

Solution:

This problem is actually one of my original problems that I made while I was riding a jeepney on my way home. I used this problem during the 9th EMC Math Grand Prix last March 1, 2014 in the Facebook group of Elite Math Circle.

The first thing you’ll notice is that there is surely a quantity that can relate the measurements of each circle of every size. You have two options – it is either a common difference, or a common ratio, but for those with very good eyesight and lightning speed thinking, they will automatically deduce that the radii of the circles has a common ratio. How did they do this?

First, let us relate the radius of the largest circle to the second circle by using one of the quadrants of the largest circle. Refer to the figure shown:

elis6 elis5

The Red Line (R) refers to the radius of the largest circle and the Yellow Line (Y) refers to the radius of the second largest circle. The Orange Line (O) is the line connecting the centers of the largest circle and the second largest circle. Notice how the orange line and the yellow line when put together forms a red line.

Geometrically speaking,

(1)O^2=Y^2+Y^2, and (2) Y=O+R

From (1),O=Y\sqrt{2}

Substituting to (2), Y+Y\sqrt{2}=R

So the ratio of the radius of the second largest circle to the radius of the largest circle is

\displaystyle\frac{Y}{R}=\displaystyle\frac{1}{1+\sqrt{2}}

Since all circles follows the same pattern, the same ratio shall govern.

If the radii are in ratio, then the area must be also in ratio. So, the ratio between the area of the circles is

\displaystyle\frac{A_{second largest}}{A_{largest}}=(\displaystyle\frac{1}{1+\sqrt{2}})^2=3-2\sqrt{2}

Going back to the problem, we are asked to find the total area of all these circles. We have a common ratio and infinite circles, so we can use the famous sum of infinite terms in geometric sequence.

S=\displaystyle\frac{a_1}{1-r}

S=\displaystyle\frac{\pi}{1-(3-2\sqrt{2})}

HOLD UP! Didn’t you notice anything fishy?

The substitution is correct if each circle of each size is only present once. That means we have to consider the number of circles present of each size. Luckily, the number of circles in each size is also in geometric progression. As seen on the figure, the number of circles in each size multiplies by 4. So correcting the substitution, we have:

S=\displaystyle\frac{\pi}{1-4(3-2\sqrt{2})}

Simplifying,

S=\displaystyle\frac{\pi}{8\sqrt{2}-11} sq. units

By rationalizing the denominator,

S=\displaystyle\frac{(11+8\sqrt{2})\pi}{7} sq. units

 

 

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