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I thought that all shapes possess translational symmetry, as simply moving it shouldn't change the shape. Then I thought of a situation where it does, though it contains three criteria.

1. The observer is fixed.
2. The shape does not possess reflective symmetry.
3. The shape is moved on a curved surface.

An easy-to-visualize example of this would be a moon shape being moved on the surface of a sphere. If it was moved halfway around the sphere, it would be flipped from the perspective of the observer (this is assuming the observer can see through the sphere).

However, is this a valid situation? If not, what examples are there of shapes that do not have translational symmetry? I cannot see how there can ever be an example of a shape without translational symmetry on a flat plane.

EDIT: This question is based on a misunderstanding of translational symmetry, my understanding coming from this site:

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1 Answer

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Usually, "translation" refers to Euclidean spaces like $\Bbb R^2$ (plane) or $\Bbb R^3$, the 3-space.

"Translational symmetry" then means you can move an object, and it still looks the same. Moving a moon across a plane or across a sphere$^1$, the moon looks different: it's in a different place and you can tell apart "before" and "after" the move (except the move was trivial, i.e. a no-move).

An example of an "object" that has translational symmetry is the set of intergers in $\Bbb R$: Moving all integers one unit to the right, you still get all integers and the set remains unchanged. The natural numbers, however, are not invariant under translation.

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Notes

$^1$Defining translation in other spaces is a bit tricky, because it should be possible to move objects around to start with. For example, you cannot move a piece of flat fabric that's flexible but does not stretch, from a flat part of a space to a place where the space is bumpy: The fabric will wrinkle and the concept of "moving objects around" will fall apart.

You can still extend the concept to other spaces, though, provided the space has constant (non-changing) curvature everywhere. Examples are

• The Euclidean space (flat, infinite in size).
• The Hyperbolic plane and space can be assigned a constant negative curvature (infinite).
• The n-Sphere with constant positive curvature (finite).
• n-Torus (flat, finite), however there is no nice isometric embedding in 3-space. Common example of a flat 2-torus is a flat computer screen where a player enters the screen again at the bottom when he leaves it at the top etc.
• Finite hyperbolic spaces with constant negative curvature like a bretzel (finite). Same problem with isometric embeddings like with the n-torus.
• Specific combinations of the objects above. For example the surface of a cylinder (flat, infinite).

Examples of translational invariant objects in hyperbolic space are hyperbolic tilings.

Examples of translational invariant objects on a 2-sphere are implied by the Platonic solids ("blow" them up so the faces lie on a sphere) and also by tilings that have $m$-fold rotational symmetry like a latitudes + longitudes coordinate grid.

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