Mathematics : Apollonius Circle applied to Tokyo Skytree and the Tokyo Tower

Mathematics class meant depression for me in school and I think am not alone in saying this. A class full of students vying for the teachers attention and the brightest in the class always miles ahead, I sank into my own space, building a gradual dislike for the subject due to peer pressure. Mathematics even today nauseates me when taught as a set of rules, defined steps and standardized questions, with very less emphasis on discovery based learning.

My tuition teacher in grade 9 was the best thing to happen to me, her emphasis was on real world problems and gave me ample time to think through problems and come with my own way to solve a particular question. Since exact steps matter a lot for “marks”, she made sure I stayed within the boundaries too. Even today, I do not just absorb objective theorems but put a subjective story to it. Stories create experiences and they help me remember the logic and purpose without making an effort at memorizing things. As an example, we all studied Apollonius Circle in school/college. How many of us remember it and when and where do we use it? So I put a subjective story to it.

STORY :  Tokyo has two towers Skytree and the Tokyo Tower. Find a location to take a snap of the Tokyo Skytree and the Tokyo Tower in one single photograph PLUS find locations in Tokyo from where both the towers will APPEAR to be of the same height. So the problem looks like the below diagramApollonius Circle

So lets take a quick look at the data. Tokyo Skytree came up in 2011 and boasts a height of 634 meters. Tokyo Tower has a modest 333 meters which is ROUGHLY the HALF of Tokyo Skytree. The two towers are separated by 8.2 kilometers.Apollonius Circle META DATA
IF Skytree is roughly 2 times the height of Tokyo Tower, then if I choose a location which is at a distance “X” from Tokyo Tower which is also at the same time “2 times X” then the relative heights of the two towers will look to be the same. Essentially what we are saying is that the proportion between the towers is 1:2, then if the location of photography is 2:1 w.r.t the towers then we should be able to achieve our aim.

Apollonius Circle as per WIKI : A circle is usually defined as the set of points P at a given distance r (the circle’s radius) from a given point (the circle’s center). However, there are other, equivalent definitions of a circle. Apollonius discovered that a circle could also be defined as the set of points P that have a given ratio of distances k = d1/d2 to two given FIXED points which is as in belowApollonius CircleDoing some calculation and with Google Maps, I zeroed on one location where I just might get that view. Ebisu Garden Place Observation Deck. Not exact but Roughly 4-5 kms from Tokyo Tower and 10-10.5 kms from Tokyo Skytree.

At Ebisu Garden Place there is a free observation tower on floor 39. There is a Chinese restaurant,Toh Ten Koh, and I booked it to check out the view. The day was clear and when I visited the observation deck, Mount Fuji was visible in the distance at the sunset time.
Ebisu Garden PlaceThe Shinjuku area in the distance also was awesome to observe from high up there. I visited here first in 2001 and it was 10 years after that I went to check the new arrival of Skytree comparing to the Tokyo Tower.
Ebisu Garden Place Time for my reservation, I went on to the restaurant and got a window seat (had booked accordingly) and behold, although I cannot say SAME height, they looked NEARLY equal as in the image below! If you need mathematical precision you can try Odaiba or the 東京工業大学 Tokyo Kougyou Daigaku from where you will get exact matching results.

Apollonius Circle applied to my hobby photography makes it easy for me to never forget the logic and provides me enough locations across the city to take similar shots! Makes sense?
Apollonius Circle

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