The Calculus of Explanations

Putting the fun in functionals

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  • The longest arc: Part II

    Last time we saw that the problem of finding the circle with the longest arc-length wasn’t possible analytically, and we had to resort to numerical or graphical methods. Now we extend the problem to find the sphere centered at the origin with the largest surface area inside a unit sphere shifted vertically by one unit. Continue reading

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  • The longest arc: Part I

    Another great problem I saw on Facebook – picture a unit circle (radius ) centred at . Which circle centred at the origin has the largest arc-length inside the unit circle? Consider a unit circle centred at , and a generic circle centred at the origin of radius , . Since the first expands to Continue reading

  • The largest area: Part III

    In Part II of this series, we wondered whether the result of the area being constant continued into higher dimensions, but, assuming it does, we also wondered whether there was any logic to the sequence of the actual value of these constant volumes. A reminder that we currently have This bugged me for a little Continue reading

  • The largest area: Part II

    Here we will consider the 3D analogue of the problem mentioned in the previous post, and first define the two paraboloids (the 3D analogue of parabolas): This time we’ll call the tangent point . Then the slope of the tangent plane of at in each direction is given by the partial derivatives of : And Continue reading

  • The largest area: Part I

    This adventure started when I attempted a harmless looking problem from the Facebook group “Actually Good Maths Problems”. The answer, which any motivated high school calculus student could find, is a sort of joke or prank on the solver, and I enjoyed it enough that I sent the problem to my good friend David Galea. Continue reading

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  • A journey into Circle Inversion

    A few years ago I watched the brilliant Numberphile video with Simon Pampena about using circle inversion to solve a problem involving Pappus’ chains, also known as Appolonian gaskets. Although the presentation was compelling and the problem interesting, it was a bit complicated and long for a first introduction to the topic. I started playing Continue reading

  • Moments of Inertia – Part V

    A slightly impractical but fascinating use of the self-similarity method is finding the Moment of Inertia of certain famous fractals. Example – The Sierpiński Triangle Consider that the Sierpiński triangle of mass and side length is comprised of three smaller copies of itself. If the triangle is being rotated about its centre, we simply need to calculate Continue reading

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