Mathematical Countryside.
List of Pages.


by 'Literka'.


This page is sponsored by Professor David McKinnon of the University of Waterloo. His main interests: number theory and geometry.




Elementary Mathematics

Monotonic subsequences. Erdos theorem.
Analysis of an elementary theorem about finite sequences of real numbers. If a finite sequence of real numbers has sufficiently many elements, than we can find a monotonic subsequence of a given length. See exact numbers and how this theorem is strengthened and generalized.

An elementary problem can be unsolvable.
This is an old problem, but still unsolved. A curious feature is that even young high school students can understand it (and they can try to solve it).


Geometric Constructions of Regular Polygons

Construction of a regular pentagon.
There are not many regular polygons we can construct with a ruler and compass. Regular pentagon is constructible, but some considerations show that it is a bridge between possibility and impossibility. Literka shows how to construct it and writes what Literka does not like in this construction.

Construction of a regular heptadecagon.
Regular heptadecagon (regular polygon with 17 sides) is also constructible, which was shown by Gauss. See the construction described on this page. See how you can use software of Literka for this construction.

Construction of a regular polygon with 257 sides.
Another constructible polygon is a regular polygon with 257 sides. Construction with ruler and compass is not so easy for this case. Description of a construction is supported by a software of Literka. Reading source files of Literka you can better understand the construction steps.



Exact Values of Trigonometric Functions

Exact values of trigonometric functions of angles (n*pi)/7.
Formulas for exact values of cos(2*pi*k/7) are given. These values are the roots of certain cubic equation.

Exact values of trigonometric functions of angles 2*k*pi/11.
Formulas for exact values of cos(2*pi*k/11) are given. These values are the roots of certain polynomial of fifth degree. Knowing that the Galois group of this polynomial is a cyclic group, we find these roots.

Exact values of cos(2*k*pi/17).
Formulas for exact values of cos(2*pi*k/17) are given. These values are the roots of an 8-th degree equation with rational coefficient.

Exact values of trigonometric functions of angles (n*pi)/17.
This page is intended for those, who are really interested in algebra, trigonometry and geometrical calculations. Seeing long formulas you will appreciate a great job that Gauss has done.

Exact values of trigonometric functions of angles (n*pi)/13.
Formulas for exact values of cos(2*pi*k/7) are given. These values are the roots of certain cubic equation.

Values of trigonometric functions of angles (n*pi)/257. Part I.
This is an introduction to a problem of finding exact values and formulas for trigonometric function of angles n*pi/257. (Number 257 is a Fermat number playing an important role in trigonometry.) Two equalities are found.

Part II., Part III, Part IV, Part V, Part VI, Part VII. Values of trigonometric functions of angles (n*pi)/257.
 More formulas of the type as in the previous link (Part I).

Values of trigonometric functions of angles (n*pi)/65537. Part I.
This is an introduction to a problem of finding exact values and formulas for trigonometric function of angles n*pi/65537. (Number 65537 is a Fermat number playing an important role in trigonometry.) Two equalities are found.

Values of trigonometric functions of angles (n*pi)/65537. Part II, Part III, Part IV and Part V, Part VI, Part VII, Part VIII, Part IX, Part X, Part XI, Part XII, Part XIII, Part XIV.
More formulas of the type as in the previous link (Part I).


Polynomials

Factorization of a polynomial, which defines values of sine function (angles n*pi/17).
This page shows how to deduce the exact values of trigonometric functions of angles n*pi/17. However, the most important are consequences of this deduction: important equalities follow.

Factorization of polynomials with roots cos(2*k*pi/n), where n is Fermat number.
When n is a Fermat number, then polynomials of the previously mentioned page can be factorized in a nice way. We can find even a factorization of a polynomial of 128 degree. Two examples are shown for n=17 and n=257.

Polynomials with roots cos(2*k*pi/n).
Take numbers cos(2*k*pi/n), where n is a fixed odd number and k=1,2,3..(n-1)/2. It appears that these numbers are roots of a polynomial with rational coefficient. Examples are shown for various n, among them for n=257. Some comments. This page is supposed to be a basis for the future considerations.


Mathematical Analysis

A Remarkable Monotonic Property of the Gamma Function.
Generalization of a result of U. Haagerup. Property of important, but still not very well known gamma function of Euler. Proof of this is so simple that perfectly fits for the internet.

Weierstrass Approximation Theorem. Bernstein's Polynomials.
See how a continuous function can be approximated by polynomials. Proof given by Literka, in our opinion, gives more intuition than analogous proof using ideas of probability.

Rudin's Theorem of Complex Analysis.
Beautiful theorem with a straightforward proof. Such theorems are very hard to invent and this rises theirs values.


Algebra

Roots of cubic equations. Cardano's formula.
Usually proofs of Cardano's formula for a root of cubic equation are not readable. Probably these proofs were not changed beginning from the time, when they were proved. Literka shows that it can be short and understandable. Literka uses a trick, but because of it you will not waste your time on reading a lot of calculations.


Roots of cubic equations. Another approach.
Another page about deriving Cardano's formulas for a cubic equation.


Roots of quartic equations. Cardano's formula.
Formulas of Cardano for quartic equations are derived.






There is a free to download software about algebraic equations. Click on Cardano's formulas.



Applications of Physics in Geometry

Weight centers of simple geometrical figures.
Weight centers are not used for proofs of mathematical theorems. Literka shows that using them can simplify many considerations.





Probability.

Positive random walks.
Probability is found that a player will play n games, under some assumptions and conditions.








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Software that you really need.


Return to the list of pages of 'Literka' about polytopes.
Return to the main geometrical page of 'Literka'.
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