When The Shape Of An Ellipse Can Be Super

Listen to Kate read: When The Shape Of An Ellipse Can Be Super

Perhaps you’ve heard that math is my everything 😁 but how wild is it that my everything is woven into, well, every thing. Like, interlaced and coming ‘atcha from all directions (though in a really non-intrusive and infinitely gentle way). I especially get down with the math when it weaves into the real world as a tool, particularly as a magic communication tool.

I’m also a big fan of the constants in life, those things that you can count on to set that bar of consistency, that baseline of composure, which may even offer a bit of predictability as you go about your day. I’m talking about knowing that the sun will rise and the sun will set at a particular rhythm through the days, months, years. I’m talking about knowing – as long as my knees keep holding up – that running any of the public staircases that are prevalent in the hills of Portland are gonna make me feel gooood (though sore, but it’s totally the good kind of sore) in the days that immediately follow that sort of exercise/self-torture.

Some mathematical constants arrive in the realm of shapes and measurement and with them they bring continuously discoverable joy to us humans in the real world. Consistently and dynamically.

Take astronomy.

Planets circling orbiting the stars, where the shape of an orbit is actually an ellipse, not a circle.

You see, you know that you’re looking at a circle when you can put a location point right in the middle of it and from that point to the line forming the circle’s shape, no matter which direction you go in, the measurement of that distance will be constant.

With an ellipse, if you go from the point in the presumed middle of its’ shape, it will not be the same measurement to the shape’s outer line.

Resembling an oval more than a circle, an ellipse has two points of reference, two points to measure from, that define its shape. Those points are known as their foci.

…the ancient Greeks had defined ellipses as the oval-shaped curves formed by cutting through a cone with a plane at a shallow angle, less steep than the slope of the conical surface itself. If the tilt of the cutting plane is shallow, the resulting ellipse is almost circular. At the other extreme, if the tilt of the plane is only slightly less than the tilt of the conical surface, the ellipse is very long and thin, like the shape of a cigar. If you adjust the tilt of the plane, an ellipse can be morphed from very round to very squashed or anywhere in between.

Infinite Powers: How Calculus Reveals The Secrets of the Universe” by Steven Strogatz
(pg. 81)

If you’re not quite picturing the whole conical shape perpendicular’d by a plane, imagine this: Shine the beam of a flashlight onto a wall. That beam is your conical shape and the wall is your plane. Then, tilt that flashlight at an angle to the wall till the beam’s shape on the plane is more oval than circular. Shahpow! You’re looking at an ellipse.

e-‘lips (noun)
A closed oval curve obtained when the plane meets only one half of the cone and its combined distance from two given points is constant.

‘fō-sī (noun)
Two distinct points of focus related to the construction and properties of conic sections.

Mathematician Johannes Kepler graced us with the ellipse and the foci. It was Kepler who legit disproved the mathematicians before him, the ones who looked upwards into the cosmos just like he did, guys like Aristotle and Galileo.

I ask you to please take a moment to imagine what that whole situation looked like: Have you ever daydreamed of disproving a hereto considered ‘expert’ that is in the same field that you are in? I imagine the giddiness in feeling that you’re about to say or do something b i g is like that moment before breaking a world record, or receiving confirmation that you’ve created something – like… let’s say refrigeration – that then becomes part of the fabric of our lives, everyone’s lives, forever eva.

With the help of tables that detailed the planets’ exact positions, Kepler later convinced himself that his [initial] model was wrong, and concluded that the planets move in ellipses, not circles, around the Sun. His new idea was promptly met with disapproval; he had failed to meet the aesthetic standard of the time.
He received criticism in particular from Galileo Galilei (1564 – 1641), who believed that “only circular motion can naturally suit bodies which are integral parts of the universe as constituted in the best arrangement.”

The Platonic solids that Kepler used to calculate planetary orbits… may be the best-known example for the conflict between aesthetic ideals and facts.

Lost in Math: How Beauty Leads Physics Astray” by Sabine Hossenfelder
(pg. 18 & 29)

Honestly, it’s this chutzpah of Kepler that I keep in the forefront of my mind when I’m in any meeting where I am holding the minority of opinion in that room yet my gut is screaming at me to hold true to my instincts and strongly vocalize – with logic and reason, of course – why things should be as I’m proclaiming them to be.

Six hundred years ago, during the period in art history known as the (European) Renaissance, the ellipse was considered to be the perfect form, used like a blueprint for painters to create a proper perspective of the composition of a scene. But then there was an even greater realization; it’s like the ellipse went from the 2D version of the best way to visually represent a gathering of people to then becoming the 3D version – the IRL version – of the best way to not only show, but to actually functionally enable, a gathering of people.

Circles. Squares. Even the almighty triangle. Sure – those are shapes that folks might sit at or gather round and congregate. Like, at a table (I’m sure there must be triangular tables, right)? And, hello, how about at the town square? I mean, right there in the name: town square. So many places of worship are modeled after theaters. Theater in the Round is a little closer. ButbutBUT, how about a town ellipse?

And this time I do not mean your basic ellipse. This time, I’m talking about a ✨SUPERELLIPSE

Piet Hein’s Superellipse
Image from GLE Superforge

Of the different shapes formed by a plane intersecting with a conical (parabolas, hyperbolas and ok fine sure, the circle too) only the ellipse can graduate to the level of super. Why? Cuz it is an intermediate of others – one that serves as a conduit of congregating.

Enter: Piet Hein, a Danish poet, painter and mathematician. In the 1950’s he was approached by a group of Swedes to help them rectify a geometrical design problem that they were having: What is the best shape for a roundabout within a rectangular space? Using simple mathematics, Hein landed on a shape that was halfway between an ellipse and a rectangle. What to call this shape? Well, taking a cue from Tina Turner, he embraced that this shape was simply the best, better than all the rest, better than any (other shape), and using the latin meaning of the word “super“- something that is above, over, or beyond, – Hein anointed this shape to be a superellipse.

More than just an elegant piece of maths, Piet Hein’s superellipse touched on a deeper human theme – the ever-present conflict in our surroundings between circles and straight lines. As he wrote, ‘In the whole pattern of civilization, there have been two tendencies one toward straight lines and rectangular patterns and one toward circular lines.’ His piece continued, ‘ There are reasons, mechanical and psychological, for both tendencies. Things made with straight lines fit well together and save space. And we can move easily – physically or mentally – around things made with round lines. But we are in a straightjacket, having to accept one or the other, when often some intermediate form would be better. The superellipse solved the problem. It is neither round nor rectangular, but in between. Yet it is fixed, it is definite – it has a unity.

Alex’s Adventures in Numberland: Dispatches From
The Wonderful World of Mathematics
” by Alex Bellos
(pg. 208)

I think we all need a town superellipse. 🤩 Efficiency in unity may just bring about the ideals of communication.

Okay — I’m now taking you with me all the way up the top step where we touch the ceiling. This ceiling-touching moment is incredibly appropriate since I’m quite stoked to share the most special use of a superellipse, and that’s when it’s used in an architectural design above our heads. It’s this expression of the superellipse that embodies that constant that I find the greatest joy in, one that is full of discovery. Welcome to a Whispering Gallery.

I lurrrve to take folks to the Whispering Gallery at Grand Central Station in NYC.
image source

The reason the acoustics are so unusual within a superellipse-shaped archway’d ceiling is because of the placement of its two foci points. When you stand at one foci and whisper into the concavity that’s in front of you and the person at the other foci has their eyes closed, they’ll swear that you must have transported directly next to them and are whispering your sweet nothings from no more than an inch away from their ear. You’ll also both look like you’re recreating a scene from some urban version of the The Blair Witch Project, but that’s besides the point.

Whispering galleries can be a romantic rendezvous – a place for lovers to covertly converse and perhaps even to commit to one another. Or these ceilings can simply be a convoy of communication – a place where banal bargaining by pompous politicians is overheard. It is believed that in Washington D.C.’s National Statuary Hall, the superellipse shape of its ceiling allowed John Quincy Adams to eavesdrop on many a conversation.

All of this deliciousness brought to you by math. Consistently and dynamically.

Thanks math, you’re the best.


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  • Let me try again: One definition of ellipse is a fixed sum of distances to the foci. Move the foci towards being infinitely far apart. Make the sum of distances only infinitesimally greater than the distance between the foci. The shape of the ellipse moves towards a straight line segment. But as a plane through a cone moves towards being parallel with the angle of the cone the shapes moves towards a parabola. Does this have more to do with the angle of the cone?

    • First off, I really like that definition of ellipse that you shared – especially the phrase / concept of ‘fixed sum’. Secondly, I’m going to make an educated guess to answer your question bc I’m at a different location than my trusty math books. I think your assumption is spot on; at first I was imagining the plane being close-to-parallel with the midpoint ‘through line’ of the cone, and I wasn’t able to match-up with what you were describing. But, yeah, as the plane becomes closer to parallel with the angle of the cone – I think of it like a dissipating run-off from the certain strength of that midpoint through-line – then yeah, that’s when a parabola makes an appearance.

      Hrmmm, I don’t think there is a way to post images in these comments as a support to your description and my description. That’d be helpful!

      Oh, and yes, as referenced in the Strogtaz book quote, an ellipse can totally take on the form as a really thin cigar — and if it was long in length, I could be convinced that it’d be ok to acknowledge it as a straight line.

    • Ooooh, thank you for posting this thoughtful question. My short answer is: Sure? Or, probably? I’m gonna reply to your expanded question on this with an expanded answer.

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