As a child, I remember being mystified how a circle could hold any great secrets. After all circles are so easy to construct and they look the same from all directions.

How wrong I was!

In studying circles, as with investigating anything of importance, one question leads to another, and insights gained in other areas of mathematics prove to be relevant to our study and reveal yet more hidden patterns. Entire books have been written about the geometry of circles, and even more books about π, the ratio between a circle’s circumference (perimeter) and diameter.

I remember being astounded when I learned that the size of the angle at the centre of a circle (the yellow angle in this image) is always exactly twice the size of ANY blue angle on the circumference! The ratio was always two, and the conclusion was that it did not matter where on the circumference you placed the angle, it would *always* be the same size … if it was standing on the same arc. Such amazing simplicity! And so, my adventure with circles began.

As you will see, there is so very much more to discover …

I have been watching your “Integration technique” videos with high expectation and excitement. They never fail to give me a lot of joy and much pleasure. You must have been one hell of teacher at your time! … just continue making highly enjoyable and intellectually stimulating videos. … Unfortunately, what you learn at the university, you hardly ever see again in your office job. That is why I try to repeat some thing I learned: to enjoy it a bit more, to marvel upon beauty, and to keep my brain sharp. There is where your videos come in perfectly: interesting, amusing and thought provoking problems neatly presented. … You do your best to share your passion, knowledge and experience. The least I can do is to say an honest “Thank you!” As I said, your videos give me much pleasure. You may freely let other people know that. … Rest assured that I shall be watching your videos in the future. With joy and excitement. And I shall comment on them from time to time. You just continue making them.

Zoran V (in private correspondence via YouTube, quoted with permission)

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