Constructing a Four-Point Egg

Do schools even teach paper/pencil/ruler/compass any more? I get not teaching slide rule, but using a straight edge and compass is even older than slide rule. I actually enjoyed drawing things like this, but I know that's not something all do. I can easily see where this is something lost with CAD and other graphing apps.

kbelder• 2 years ago

That's important, I think... not just historically, but to assist in concept formation. Although, replace 'ruler' with 'straightedge', to emphasize that there is no measuring.

shashanoid• 2 years ago

Thanks having fun on geogebra

rantallion3• 2 years ago

What makes this a four-point egg if only the first three points are required to derive the rest of the shape?

burkaman• 2 years ago

See Figure 15 on page 8: https://web.archive.org/web/20200618202007/https://www.dynam...

The egg is constructed of four distinct arcs based on four different points, plus two of them mirrored.

rantallion3• 2 years ago

Thanks. This makes more sense. I was curious why the _inner_ point mattered, when the fourth point is actually the one on the side.

a_t48• 2 years ago

There are four points inside the egg.

kazinator4• 2 years ago

Also, why do we say "four legged table" in reference to common, rectangular tables? The position of the fourth leg is determined from the other three, so it shouldn't count!

rantallion3• 2 years ago

I'd love to see a three-legged table stand up with the fourth corner under load.

dylan6044• 2 years ago

why do we say a rectangle has four sides, when it's just 2 reflected

gowld• 2 years ago

The Wikipedia article about the 2-point (Moss) egg (3 points derived from 2 freely chosen points) cites Freyja's article https://web.archive.org/web/20200618202007/https://www.dynam...

which has a 2-point (Moss) egg, a 4-point egg, and a 5-point egg. The OP article has only the 4-point egg.

fanf23• 2 years ago

Moss’s egg is a 3 point egg: it has 4 arcs, two of which are a mirrored pair.

SamBam2• 2 years ago

Only the very first circle is needed to derive the rest of the shape. So that's a point and a radius, or two points.

nick238• 2 years ago

Two points (x1, y1, x2, y2) or 4 total parameters (x, y, rotation, scale) assuming you're using the exact shape provided, but from the interactive egg you can free the location/angles that they used for construction so you can wind up with less egg-like 4-point eggs.

In the interactive link provided, you control the positions of 4 circle centers, each restricted along one of the axes. So you're giving 4 parameters (which also gives scale), and after including x-y offsets and rotation, it's 7 total numbers to make an arbitrary '4-point' egg anywhere in the plane

lupire3• 2 years ago

That's for a specific case of 4-point egg. The points can be moved to make different egg shapes. The OP links to an interactive app.

samstave• 2 years ago

It looks like the four-point egg is defined by 8 circles, and connecting the intersections, of which their are four pair.

Im just surprised the author never once used the proper term for the "west" "north" points; Quadrant. Its even the name for the snap function in all snap menus.

fanf23• 2 years ago

A quadrant is a quarter of a planar shape, not a point.

etwigg• 2 years ago

I didn't know an egg could be so beautiful <3

1970-01-01• 2 years ago

Eggcellent :)

quickthrower2• 2 years ago

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ks2048• 2 years ago

Nice. I wonder how closely it could be approximated with a small number of Bezier curves. (for example, a quadratic with six points. 4 control points and two side points).

ryandrake• 2 years ago

I wonder if you could make it even more "beautiful" by gradually easing between the various curvatures rather than having the discontinuities at the intersections of the curves. Kind of like the difference between a square with rounded corners and a "squircle"[1].

1: https://webflow.com/blog/squircle-vs-rounded-squares

alexchamberlain• 2 years ago

Do you have discontinuities though? The tangents at the point of intersection match.

fanf23• 2 years ago

There is a discontinuity in curvature where each arc meets. It’s possible to make joins between curves smoother using Euler spirals, but I don’t know how to use them well enough to apply them to eggs. https://en.m.wikipedia.org/wiki/Euler_spiral

I recently tried to draw f holes using Euler spirals, but they turned out too short and fat, and I gave up on them. But they are quite fun to draw on a computer by simple numerical integration, ignoring all the more complicated mathematics in the wikipedia article. Perhaps I could have made my f holes more elegant by understanding the parts I ignored… https://en.m.wikipedia.org/wiki/Sound_hole

pimlottc• 2 years ago

The issue is more the smoothness in the change in curvy-ness:

https://www.figma.com/blog/desperately-seeking-squircles/

onetimeuse92304• 2 years ago

If you sit in a rolling car and the driver suddenly presses the break vs eases it in, you could halt with the same deceleration and yet one will feel smoother than the other.

The car's position is not discontinuous, the velocity isn't either (in both cases it has to slow through intermediate velocities).

It is the decelaration that in one case grows suddenly and in the other changes smoothly.

On an egg, there is no place where curvature changes suddenly, and yet on a four point egg there are points where it does exactly that.

TeMPOraL2• 2 years ago

From [1]:

“This may seem trivial, a cool story, but subconsciously it really makes a big impact: a squircle doesn’t look like a square with surgery performed on it; it registers as an entity in its own right, like the shape of a smooth pebble in a riverbed, a unified and elemental whole.”

Am I the only one who prefers squares with rounded corners?

"Shape of a smooth pebble" is not something I like to see anywhere outside a garden. A "squircle" looks inflated, like Li-Ion battery about to pop and set your house on fire, or like package with food that spoiled, and will stink if opened, and make you sick if eaten. You need to hold your breath around a "squircle", because the moment you breathe out and time unfreezes, it'll start rolling and tumbling. Even nature itself doesn't really do "sqiurcles" unless it must. The last "squircle" I saw was when I burned my back badly and had them pop up on my skin (if you read this and think "gross", well, that's what "squircles" are for me in general).

The most benign connotation I have with a "squircle" is a pillow, which again is something that doesn't belong in almost any context.

A square with rounded corners, in contrast, looks purposeful. An object created with intent. It speaks of precision, of quality, of human workmanship. It looks stable, unspoiled, new. It fits.

blt2• 2 years ago

Here's a more analytical approach to constructing egg geometry: https://kar.kent.ac.uk/89885/

twic• 2 years ago

And a purely empirical approach: https://www.imperial.ac.uk/news/179909/how-birds-determines-...

throwaway2037• 2 years ago

Who is the "Moss" in Moss's egg? I assume it is a person, but I cannot find anything when I search.

jhardy54• 2 years ago

This led me on a bit of a rabbit hole, but I’m suspicious that Moss may refer to Stephanie Moss, who is referenced in the preface of Mathographic by Robert Dixon.

Constructing a Four-Point Egg

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