# Reflections and Reflection Groups - Basic Geometry

• Math Amateur
In summary, Kane explains that a "hyperplane" is a maximal proper subspace of a space and that a reflection can be defined with respect to this subspace. He then goes on to explain that a "reflection with respect to hyperplanes" is just a special case of a reflection with respect to a vector. Finally, he provides a summary of the rest of the chapter.
Math Amateur
Gold Member
MHB
I am reading Kane's book on Reflections and Reflection Groups and am having difficulties with some basic geometric notions - see attachment Kane paes 6-7.

On page 6 in section 1-1 Reflections and Reflection Groups (see attachment) we read:

" We can define reflections either with respect to hyperplanes or vectors. First of all, given a hyperplane $H \subset E$ through the origin, let L = the line through the origin that is orthogonal to H. So $E = H \oplus L$"

My question is why/how is $E = H \oplus L$?

Can anyone help?

(see my intuitive diagrams - my notion of $H \oplus L$ is a line going through a plane)

Maybe I need to read some basic differential geometry?

Peter

#### Attachments

• Kane - Reflection Groups and Invariant Theory - Pages 6 - 7.pdf
117.9 KB · Views: 218
• Direct Sum of H and L.pdf
16.4 KB · Views: 221
Last edited:
Math Amateur said:
I am reading Kane's book on Reflections and Reflection Groups and am having difficulties with some basic geometric notions - see attachment Kane paes 6-7.

On page 6 in section 1-1 Reflections and Reflection Groups (see attachment) we read:

" We can define reflections either with respect to hyperplanes or vectors. First of all, given a hyperplane $H \subset E$ through the origin, let L = the line through the origin that is orthogonal to H. So $E = H \oplus L$"

My question is why/how is $E = H \oplus L$?

*** Without the basic definitions I must rely on the standard ones, and then H is a hyperplane = a maximal proper subspace of E = the kernel of some non-zero lin. functional = a subspace of dimension n - 1 if dim E = n.

Thus, H is a hyperplane iff $E=H\oplus Span\{v\}$ , for any $v\notin H$ , and this is basic (not necessarily finite-dimensional) linear algebra, no differential geometry needed at all.

DonAntonio

Can anyone help?

(see my intuitive diagrams - my notion of $H \oplus L$ is a line going through a plane)

Maybe I need to read some basic differential geometry?

Peter

...

If H is a "hyperplane" in space E through the origin, E having dimension n, then H is a subspace of E of dimension n-1. There exist a basis for H consisting of n- 1vectors which can be extended to a basis for E by adding one more vector perpendicular to all the n-1 basis vectors in H. The space spanned by that one vector is L. every vector in E can then be written as a linear combination of the basis vectors for H, and so is in H, and a vector in L. That is essentially what "E= H⊕L" means.

Thanks for the help!

Peter

Hello Peter,

Thank you for sharing your question and diagrams. I can understand your confusion with the notation and definition of reflections in Kane's book. To clarify, the notation E = H \oplus L means that the vector space E is the direct sum of the two subspaces H and L. In other words, every vector in E can be uniquely expressed as a sum of a vector in H and a vector in L. This is a fundamental concept in linear algebra.

In the context of reflections, this means that any vector in the vector space E can be written as a sum of a vector in the hyperplane H and a vector in the line L. This is because reflections are defined as transformations that flip a vector across a hyperplane or a line, while keeping the hyperplane or line fixed. Therefore, the vector space E is divided into two parts - the hyperplane H and the line L - where each part represents the fixed and moving components of the reflection transformation.

I hope this explanation helps clarify the notation and concept of reflections and reflection groups in basic geometry. If you are still having difficulties, I would recommend reviewing some basic linear algebra concepts, such as vector spaces and direct sums, before continuing with Kane's book. Good luck with your studies!

## 1. What is a reflection in basic geometry?

A reflection is a transformation in geometry that involves flipping or mirroring an object over a line of reflection. This line acts as a mirror, so the image will be a mirror image of the original shape. Reflections preserve the size and shape of the object, but its orientation is reversed.

## 2. How do you perform a reflection in basic geometry?

To perform a reflection in basic geometry, you need to follow these steps:

• Draw the line of reflection, which will act as the mirror.
• Identify the points of the shape that are equidistant from the line of reflection. These points will be the same distance from the line on both sides.
• Draw lines connecting each point to its reflection across the line of reflection. These lines should be perpendicular to the line of reflection.
• The reflected shape will be the mirror image of the original shape.

## 3. What is a reflection group in basic geometry?

A reflection group is a group of reflections that can be combined to create different geometric transformations, such as rotations and translations. These groups are used in symmetry and pattern recognition in mathematics and art.

## 4. What are some examples of reflection groups in basic geometry?

Some common examples of reflection groups in basic geometry include:

• The group of reflections that form a square, which can be combined to create rotations and translations that preserve the square's symmetry.
• The group of reflections that form a regular hexagon, which can be combined to create rotations and translations that preserve the hexagon's symmetry.
• The group of reflections that form a kaleidoscope pattern, which can be combined to create beautiful and intricate designs.

## 5. How are reflection groups used in real life?

Reflection groups have many applications in real life, such as in crystallography, where they are used to study the symmetries of crystals. They are also used in architecture and design to create symmetrical patterns and designs. In addition, reflection groups are used in computer graphics and animation to create realistic reflections and symmetrical patterns in digital images.

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