Nice intro to connections between algebra and geometry

In summary, the conversation mentions the connections between algebraic structures and geometries, and the need for a book that explores these connections in a way that is accessible to undergraduate students. The desired book should provide thorough explanations and examples, including examples from physics, to help clarify concepts such as why geometries correspond to groups instead of fields or rings, and the implications of properties like being abelian or Lie groups for geometric properties. The article "Geometries and Groups" and the book "Glimpses of Algebra and Geometry" are recommended as potential resources for this topic.
  • #1
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Connections between algebraic structures and geometries are mentioned in almost any course of modern geometry or algebra. There are monographs dedicated to the subject. Unfortunately, the books, I managed to find, are written for professional mathematicians. I am looking for a book that focuses on those connections written with (under)graduate in mind, with thorough explanations and examples (maybe, not just purely mathematical examples, but ones borrowed from physics), that could answer my questions like the following (I know they are incorrectly posed, too vague, even naïve, but this is exactly why I am in need for such a book, to be able to ask right questions):

- why geometries correspond to groups, but not to fields or to rings (those possesses group structure anyway)?
- properties of groups to be abelian or Lie groups; what does it mean for the geometric properties?
- geometric meaning of existence of normal subgroups and quotients; zero dividers?
- complex numbers are intimately related with geometry; what about quaternions, I know they are used to describe rotations, but this seems to be scanty compared to rich geometric applications of complex numbers; why is that so?

Again, I am not looking for answers for the questions above, but for a book that would help me to dissipate darkness surrounding them.
 
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  • #3
basic fact: given a geometric space, such as a euclidean plane, the set of one to one transformations that preserve the geometric properties like distance, form a group. more interesting, given an abstract group, one can search for a geometric object whose set of transformations are identical with this group.

here is a very well written book, elementary in terms of prerequisites, but still fairly challenging:

Geometries and Groups, by Nikulin and Shafarevich.
 
  • #4
A recommended book sitting on my desk, that I have not gotten to yet:

Toth's Glimpses of Algebra and Geometry
 
  • #5
I'm grateful to everyone for the suggestions. I'll take a look at all of them.
 

1. What is the connection between algebra and geometry?

The connection between algebra and geometry lies in their shared use of variables and equations to represent and solve problems. In algebra, variables are used to represent unknown quantities and equations are used to show relationships between these quantities. In geometry, variables are used to represent the dimensions of shapes and equations are used to calculate measurements and solve for unknowns.

2. How does algebra help with understanding geometry?

Algebra helps with understanding geometry by providing a way to represent and manipulate geometric concepts using symbols and equations. For example, the Pythagorean theorem in geometry (a² + b² = c²) can be translated into algebraic form (x² + y² = z²), making it easier to solve for unknown sides or angles of a right triangle.

3. How does geometry help with understanding algebra?

Geometry helps with understanding algebra by providing visual representations of abstract concepts. For example, geometric shapes can be used to illustrate the concept of variables and how they can change in relation to each other in an equation. This can help students better understand the concept of solving equations.

4. What are some real-world applications of the connection between algebra and geometry?

The connection between algebra and geometry has many real-world applications, such as in architecture, engineering, and computer graphics. In these fields, geometric principles and equations are used to design structures and create visual representations of objects, while algebra is used to solve problems and make calculations.

5. How can understanding the connection between algebra and geometry improve problem-solving skills?

Understanding the connection between algebra and geometry can improve problem-solving skills by providing a more comprehensive approach to solving problems. By using both algebraic and geometric methods, students can gain a deeper understanding of the problem and develop critical thinking skills in finding solutions. Additionally, the ability to translate between algebraic and geometric representations can make problem-solving more efficient and effective.

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