Wednesday, June 22, 2016

Ceva's theorem

Simple yet powerful theorem that gives the properties of concurrent lines from vertices in a triangle is Ceva's theorem. If the lines AD, BE and CF are concurrent, then  AF/FB x BD/CD x CE/EA = 1. This is attributed to Italina mathematician Giovanni Ceva.

The lines that start from vertices to opposite edges are called cevians. Some examples are medians, angle bisectors, altitudes. Theorem can be used to easily determine whether three such lines are concurrent. Intersection points of such lines are significant centers of triangle in geometry. Some example centers of triangle are centroid, incenter, circumcenter and orthocenter.

* Ceva's theorem
* Ceva's theorem
* Triangle Centers

Tuesday, June 21, 2016

Burnside Lemma

Group theory is formal study for analyzing systems and processes that have symmetry and structure. When an operation is performed on some element, it preserves some structure. Many surprising patterns and discoveries can be made about such symmetries and structures.

Groups are applicable in fields like polynomial equations, polymer structures, topology, number theory, probability, quantum physics and combinatorics. Cauchy, Galois, Cayley, Frobenius, Polya, de Brujin, Redfield and and other several mathematicians contributed to field, but Burnside collected all research and publicized it in his book Theory of groups of finite order in addition to his own contributions to the field.

Group is mathematical abstraction that consists of a set of elements and an operation that satisfies certain properties on the given set. Set of integers with addition form a group. The theory will be more beautiful in dealing with sets of finite order and their symmetry. Set of rotations of rubic cube with combining operation form a group. Symmetry group or permutation group is group whose set is set of transformations like rotation, reflection and moving of an object and whose operation is composition of transformations.

Burnside lemma is a result in group theory that says the number of orbits of input set operated under a group of transformations is equal to the average number of points fixed by the transformations of that symmetry group. It is used to count distinct possible objects or configurations considering the symmetry of objects.

Each possible configuration is an input point. Orbit of a point is set of all possible points possible by applying a transformation of group. Orbits are subset of all points. All elements of orbit share the orbit. Orbits do not overlap and they partition the set of input configurations. Fixed points of a transformation are unchanged points after applying it. Many points may be identical and are simply few transformation away from other points and are so become one orbit. The number of orbits are the number of distinct possible objects or configurations.

Example: Counting the number of bracelets with total four beads of two colors. Two bracelets are considered same if they look identical after some rotation or flipping. There are eight symmetries.
  • No movement will keep all sixteen bracelets intact.
  • Two bracelets do not change after rotation by one bead.
  • Four of them do not change to original after rotation by two beads.
  • Two of them do not change to original after rotation by three beads.
  • Four of them do not change after horizontal flip.
  • Four of them do not change after vertical flip.
  • Eight of them do not change after clockwise diagonal shift.
  • Eight of them do not change after anticlockwise diagonal shift.
  • Total fixed bracelets is  48, that is 16 + 2 + 4 + 2 + 4 + 4 + 8 + 8.
  • Total distinct number of bracelets is 48/8 or six.

Sunday, June 12, 2016


Six sigma is set of techniques and tools to maintain quality of manufacturing or serving processes in industry. It was developed at Motorolla but was later adopted by General Electric and other companies.

The Six Sigma follows two processes.
  • DMAIC for existing processes - define, measure, analyze, improve and control
  • DMADV for new process - define, measure, analyze, design and verify.
Sigma is statistical term that measures how far something deviates from average. Six sigma process means a process that has no more than 3.4 defects per million oppurtunities. The idea behind six sigma is that you can measure how many defects in your process and figure out how to eliminate them.


Friday, June 10, 2016


One of the great geniuses ever lived is Leonhard Euler. He contributed to number theory, topology, calculus, infinite series and many branches of the mathematics in 18th century. He started his education in Basel and worked in St. Petersburg and Berlin for most of his life.

Some of the discoveries by Euler
  • e^ix = cos(x) + i sin(x) and e^i(pi) + 1 = 0
  • v - e + f = 2 to describe relation between verities, edges and faces of polyhydra. 
  • Euler line on which orthocenter, circumcenter and centroid are collinear for any triangle.
  • Euler totient function to count the numbers that are relatively prime to a number
  • Summation of series like ∑ (1/n2) , ∑ (1/n!), ...

Sunday, June 5, 2016

Problem solving

Problem solving strategies - Arthur Engel
Mathematical Problem Solving - Alan-Schoenfeld
Thinking Mathematically - J. Mason, L. Burton and K. Stacey
How to Solve it - G. Polya, John H. Conway
The Art and Craft of Problem Solving - Paul Zeitz