Constructing a Four-Point Egg

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

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.

One free tool for making such diagrams is GeoGebra.

Online: https://www.geogebra.org/geometry

Offline: https://www.geogebra.org/download

(I have no affiliation with it. It was used in an article OP's submission links to.)

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

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

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

Great egg. Thanks for posting

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

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).

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

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

Thanks having fun on geogebra

You can also try paper, pencil, ruler and compass.

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.

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

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.

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.

There are four points inside the egg.

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!

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

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.

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.

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.

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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.

wow, apparently this comment touched a nerve

You can try a bit of burnt wood and a cave wall.

I do hate when f holes are too short and fat, but yes: understanding the parts you generally ignore might help.

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.

Eggcellent :)

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

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

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

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

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

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

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.

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Yes, but they use a "non-collapsing" compass that can transmit distances without constructing them, not Euclid's compass (which can be used to transmit distances, with more work).

http://euclidea.xyz

You can use a stick and some dirt, as long as no Macedonian kings block your sun

I choose ruler over straightedge because that is what people will have more commonly and you can always ignore the scale.

i have this mechanical drafting book - it's fucking amazing. Goes from simple euclidean type stuff - all the way to full parts, amazing patterns and so on: https://www.amazon.com.au/Popular-Mechanics-Art-Mechanical-D...

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

There are ways:

(1) Diagonal legs that give the table a wide stance. Draw a triangle that the tabletop fits entirely inside of. The legs touch the ground at the vertices of the triangle.

(2) Make the table itself arbitrarily heavy. A downward force on the corner of the tabletop will create torque around an axis, and this torque wants to tip it over. But the table's center of mass is on the other side of this axis, and the table's weight creates torque too. If the table's mass is high enough, this torque is greater, and it won't tip.

(3) Screw the table to the floor. Is this a table? I think it's still a table. You often see tables attached to the floor.

(4) Make one leg really wide so that it stretches from one corner to the other. The other two legs can be traditional.

It has a leg there; it's just called a three legged table, because one of the legs is geometrically implied by the other three.

I like how you responded to a question about not using apps with an app.

I don't know exactly what you mean, but the name of this shape is consistent with the naming of traditional geometric constructions like the "five-centered arch" and the "three-centered arch" (see many diagrams online for details of these constructions).

So "four-centered egg" might make more sense to you, though it could also be called the "six-centered egg", because it includes both sides, unlike the classical arch constructions.

Surely you only need two, as you could use two diagonal legs.

Let's see if the name "two-legged table" takes off.

my grandfather was a custom cabinet builder, and for whatever reason 4 legged tables were his absolute nemesis. i have the last remaining 3 legged table of his. it's a round table, so i thought the 3 legs were a design choice. that notion was corrected when my dad started laughing at remembering his dad's sheer frustration when he realized 3 of 4 table legs were the same length...on multiple occasions.

The table + load will tip over if the center of mass is vertically outside of the convex polygon formed by the tips of the legs (in this case a triangle); i.e. the base of the object.

Center of mass over base -> stable

Center of mass outside base -> capsize

You only need 1 center leg.

Even if a four legged table's legs are all perfectly equal so their tips lie in the same plane, that perfectly will be thwarted by an uneven floor.

> Even if a four legged table's legs are all perfectly equal so their tips lie in the same plane, that perfectly will be thwarted by an uneven floor.

Unless you're able to rotate it: https://www.scientificamerican.com/article/strange-but-true-...

Wish I'd be able to rotate my floor instead.