Mathematics Stack Exchange
Mathematics Stack Exchange

Questions tagged [symmetry]

Questions about symmetry, in group theory, geometry or elsewhere in mathematics. See https://en.wikipedia.org/wiki/Symmetry

1533 questions
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Reflection with respect to a parabola

I know how to find a reflection with respect to one of the axis or with respect to the origin, but let's say I want to find the reflection with respect to a parabola, how do I do it? Let's say we have the standard $y = x^2$, in this case each point…
user
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Do symmetric 3d solid figures always have a plane of symmetry?

Chapter 3 of Martin Gardner's The New Ambidextrous Universe begins as shown below. As you can see (highlighted), on page 13 he writes that not all solid symmetric objects have a plane of symmetry, and gives an example. On the next page, however, he…
I. J. Kennedy
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1 answer

How do you prove you've found all symmetries of an object in 3D?

In group theory class we studied the example of rotational symmetries of the regular tetrahedon. The teacher showed us 12 symmetries and then said "if you stare long and hard you can convince yourself that those are all symmetries". Is there a way…
user388557
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5
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Determining the symmetry group of an infinite horizontal line.

I believe I have a satisfactory answer to the following question: Imagine we have a infinite horizontal line running through the origin, what is the associated symmetry group? Now thinking intuitively about the symmetries of this line we can see…
George1811
  • 1,981
3
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An isomorphism between the full tetrahedral symmetry group and the cubic rotation group?

I know that the full symmetry group of the tetrahedron and the rotation group of the cube are both isomorphic to $S_4$. But I want to show a direct, visual isomorphism. I tried looking at a tetrahedron inscribed in a cube, but odd rotations of the…
Nishant
  • 9,155
3
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1 answer

The intersection of two symmetry planes is a symmetry axis

I wonder if the intersection of two planes of symmetry for some three dimensional object, is a symmetry axis of that object (i.e. an axis for which there exists an angle (smaller than 360°), so that rotation around this angle maps the object in…
Rayman
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Is there a reason mathematical operations come in pairs?

Writing out the list of common mathematical transformations, there's an apparent symmetry of pairs: addition and subtraction multiplication and division powers and roots trigfunctions and arctrigfunctions matrix multiplication and…
Ben
  • 153
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1 answer

Symmetries of a Soccer Ball

The FIFA World Cup 2018 in Russia is played with an Adidas ball, called Telstar 18. It consists of 6 square panels and 12 panels with symmetry of a parallelepiped, but with curved sides. The ball has the octahedral symmetry group without…
Hans Block
  • 31
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What is the symmetry of the Penrose tiling?

What is the symmetry of the Penrose tiling? Simply C5 or bigger? Any simple proof that the tiling is a complete cover of the plane?
Beamer
  • 1,079
2
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1 answer

Isometries of the plane

Let $m$ be an orientation-reversing isometry. Prove algebraically the $m^2$ is a translation. What I attempted: We know that $m$ is an orientation-reversing isometry i.e. it is either a reflection or a glide-reflection. I started by consider the…
amir
  • 1,311
2
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1 answer

Shapes with $7$ lines of symmetry

I am trying to find shapes with $7$ lines of symmetry. Regular $7$-gon(Heptagon) has this property. but can you give example of other shapes with $7$ lines of symmetry? (I know I can draw an small circle on every vertices of the Heptagon and still…
Etemon
  • 6,437
2
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1 answer

Show that a $2D$ region is symmetric around the line $y=x$ or $y=-x$

I need to show that the following region is symmetric around the line $y=x$ and $y=-x$: $$\{(x,y)|x^2+y^2+xy\le0.5\}$$ How can I do it algebraically (without drawing the region)?
Daniel
  • 558
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Symmetry of tetrahedron that is not a reflection nor a rotation

Studying a tetrahedron I have identified twelve rotational symmetries and twelve reflectional symmetries. Now I am asked to identify a symmetry that is not a reflection nor a rotation, but which is equal to the product of three reflections. I…
H. Vabri
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Using symmetry to find the maximum of a function

Without using calculus I was told to find the maximum of $$f(x)=6x-x^2.$$ The "symmetry" approach notices that $f(x)=x(6-x)$, and that replacing $x\leftrightarrow6-x$ does not change $f(x)$ which means it doesn't change the maximum. Then the…
matryoshka
  • 981
2
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No symmetries, or $\mathbb{Z}$?

I was thinking about this question from the other day, and some good answers were provided for geometric objects with symmetry group $\mathbb{Z}$. It just occurred to me, though: why isn't $\mathbb{Z}$ the symmetry group of, say, a capital letter…
G Tony Jacobs
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