2.6 Symmetry and Unification Discussion
Discussion

We’ve seen that one object can be “more” symmetrical than another, based on the number of rotational symmetries it has. A circle has infinitely many of these symmetrical transformations, but we also learned that a sphere does as well. Can we say that one of these objects is more symmetrical than the other?

February 5, 2021 at 6:27 am
Sure. Sphere is the most symmetrical object. One can rotate it how one likes and it always remains the same.

February 9, 2021 at 3:49 am
As far as I can tell this is a massively complex topic but I would guess it is provable much like (or through the use of) Cantor’s proof of cardinality in number sets (i.e. different sizes of infinity). The sphere definitely has more degrees of freedom to play around with while generating multiple infinite symmetrical transformations so it seems intuitive that it would be a higher cardinality… not sure though, need to learn the proof.

March 2, 2021 at 10:11 pm
I believe the sphere in any dimension is the most symmetrical object in such a dimension.
Due to all spheres are isomorphic all of them has the same infinite number of symmetries. 
March 6, 2021 at 7:17 pm
I think Spheres are more symmetrical, they always are the same (talking about symmetric)

March 7, 2021 at 8:37 pm
I think the cardinality of the points of a circumference on a circle and the points on the surface of a sphere is the same because there can be a onetoone matching of one to the other. There are just as many symmetries in a circle as in a sphere; both are infinite.

March 11, 2021 at 8:54 pm
I would also answer, although I am not familiar with group theory, that a sphere is more symmetrical…But there is from another point of view an infinite number of points in a circle as in a sphere…so maybe with Cantor (?) one could say that one infinite set can be greater than another infinite set, although this is counterintuitive…

March 19, 2021 at 8:54 am
The sphere has more degrees of freedom ( three axes ) to express it’s symmetry than the circle ( one axis ), and a consequent richer complexity of it’s symmetry group.
But here lies a more general point : it is in the eye of the beholder, as witnessed by the variety of possible answers to the given question. But I personally go with Feynman. If you lose the mathematics you lose the understanding. So the group structure of the eightfold way is the minimum that you need in order to keep an intellectual handle on the particle zoo. Solving a Rubik’s Cube is harder than mere rotation of a square. Rotation of a circle invokes an uncountable number of choices compared to any approximating polygon. The mathematics can give fruitful surprises ….. so I equate greater symmetry of one object over another as the greater complexity in the underlying group that one chooses to model with. Some may prefer simplicity and that can be a valid viewpoint too.

July 16, 2021 at 2:11 pm
Of course yes, here circle is 2D but sphere is a 3D object. Hence sphere have more symmetrical transformations and we can say that sphere is more symmetrical than circle.

July 21, 2021 at 7:02 am
the sphere is more symmetrical than a circle because it’s a threedimensional object, hence, it could rotate in different directions such as X, Y, and Z regarding its axis and still preserve its shape.

October 22, 2021 at 11:52 am
The circle and the sphere both have the same symmetry. Infinity. Both are countably infinite.

February 8, 2022 at 8:59 pm
It all depends on the layer of abstraction you look at the objects around you.
A human face could be judged to be symmetrical, in alignment from one side to the other, a sphere could be symmetrical, but on the molecular level perhaps it isn’t, maybe one of the quarks isn’t stable at the quantum level. Perhaps the human’s skin isn’t as symmetrical as it appears to be.Perhaps they are both in symmetry, in the sense of coordinate plans and boundary layers. A but circle has more symmetry, if I was to fry a circle in oil and a square in oil the circle would fry evenly, whereas the square with its pointer ends would end up more crisp. The circle is thus more symmetrical. this is because of the differential of distance from one side to the other. the circle all points are equal, but the square this is not the case. There are differences in thickness.

September 16, 2022 at 8:02 am
A sphere has more sumetries than a circle.
Flat Euclideans surfaces are not closed igloos nor open saddles.
But… 🙃
Higgs is the field of symetry it is said to occur from. 🗺
When we see the Higgs particle attatch to others, almost a piggyback 🐖, there is a bonding.
Bonding is the strong nuclear force.
🔗
If Biggs bonds to decay, it is the weak magnetic force.
💦🌧🌨💧🌊❄
How can biggs be electromagnetic? It can be displaying quanta… AEther… Of both a wave and a particle duality. 🎭How can Higgs be gravitational? What is a graviton? ⚫holes❕ ⚪holes✔

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