Intereting Posts

Is there an empty set in the complement of an empty set?
How to calculate the improper integral $\int_{0}^{\infty} \log\biggl(x+\frac{1}{x}\biggr) \cdot \frac{1}{1+x^{2}} \ dx$
How to deduce open mapping theorem from closed graph theorem?
Discrete valuations of the rational numbers
Norms on C inducing the same topology as the sup norm
Are $T\mathbb{S}_2$ and $\mathbb{S}_2 \times \mathbb{R}^2$ different?
How many times to roll a die before getting two consecutive sixes?
Evaluation of the sum $\sum_{i=1}^{\lfloor na \rfloor} \left \lfloor ia \right \rfloor $
Is $\aleph_0$ the minimum infinite cardinal number in $ZF$?
Symmetric random walk and the distribution of the visits of some state
Understanding the multiplication of fractions
Pigeonhole Principle to Prove a Hamiltonian Graph
Differential Geometry without General Topology
Concerning $f(x_1, \dots , x_n)$
Which of these courses to take if one intends to go to grad school in pure math (rank please)

First of all, I want to master Geometry, I have knowledge on high school geometry and I was thinking of learning Euclidean Geometry. I bought a copy of Euclid’s Elements, it is very interesting, however, it does have a fairly different method compared to the modern approach in teaching geometry. Can I ask if it is required in our modern mathematics to learn Euclid’s Elements? Or is learning Euclid’s elements just for intellectual exercise? Are there any modern textbook on Euclidean Geometry or plane geometry? I have no problem with the formal mathematical approach using Axioms and Postulates, I enjoy having a first exposure to them, actually.

In the future, I want to read Principia Mathematica by Isaac Newton, is it a must to learn Euclid’s Elements to learn it? Or Descartes’s Geometry is the basis of it? Or maybe there is a modern geometrical approach to explain it?

- The shortest distance between any two distinct points is the line segment joining them.How can I see why this is true?
- Tangent Points to Ellipse
- Construction of a regular pentagon
- About the ratio of the areas of a convex pentagon and the inner pentagon made by the five diagonals
- New very simple golden ratio construction incorporating a triangle, square, and pentagon all with sides of equal length. Is there any prior art?
- Is there a name for the recursive incenter of the contact triangle?

- Characterization of Volumes of Lattice Cubes
- Formula for solving for Cx and Cy…
- Proving two lines trisects a line
- Can one always map a given triangle into a triangle with chosen angles by means of a parallel projection?
- What is the solution to the Dido isoperimetric problem when the length is longer than the half-circle?
- How did Beltrami show the consistency of hyperbolic geometry in his 1868 papers?
- Another chain of six circles
- How to show that any rectangle in ellipse must be oriented parallel to axes?
- Solid body rotation around 2-axes
- Why does the Pythagorean Theorem have its simple form only in Euclidean geometry?

A more ‘modern’ way to study Euclidean geometry is to recast all theorems and prove them using methods of Linear Algebra, using coordinate space R^2 and R^3.

I would be interested if there was an author who would accept this challenge. The nice thing about linear algebra is that you can verify results easily using a computer.

There are advantages and drawbacks to using Linear Algebra. In linear algebra proofs tend to be more compact and involve more algebraic type manipulation. In synthetic geometry proofs involve the use of complicated diagrams and tend to be wordy.

On the other hand, in synthetic geometry it is easier to draw such basic figures as a line segment, while in R^2 or R^3 we would have to use parametric equations. It is easier to ‘discover’ geometric relationships when you can draw lines and circles freely as we do in synthetic geometry. Still, proving these statements tends to be more compact using coordinate space or methods of Linear algebra.

Personally i’m not a fan of reading synthetic geometry proofs, that is why i am interested in a different approach to Euclidan geometry. why not have the best of both worlds, the lean compactness of linear algebra proofs but discover them using the normal euclidean tools of points, lines, circles, etc.

- If $f\in\hbox{Hom}_{\mathbb{Z}}(\prod_{i=1}^{\infty }\mathbb{Z},\mathbb{Z})$ and $f\mid_{\bigoplus_{i=1}^{\infty } \mathbb{Z}}=0$ then $f=0$.
- Are infinitesimals equal to zero?
- Why do we accept Kuratowski's definition of ordered pairs?
- Bijective mapping from $(-1,1)$ to $\Bbb R$
- Showing summation is bounded
- Show that $\int_0^\pi \log^2\left(\tan\frac{ x}{4}\right)dx=\frac{\pi^3}{4}.$
- Useful techniques of experimental mathematics (reference request)
- What is the group of units of the localization of a number field?
- Another Laplace transform of a function with square roots.
- What is the Probability that a Knight stays on chessboard after N hops?
- Efficient Algorithm for finding left (or right) Transversal in a Group
- Degree of splitting field divides n!
- Choosing $\lambda$ to yield sparse solution
- A paradox on Hilbert spaces and their duals
- Modular Inverses