Monthly Archives: August 2020

Why 7 Frieze Patterns ?

Posted on August 23, 2020 by Sumant Sumant

Frieze pattern are widely used in architecture and art. These are two dimensional repetitive patterns in one direction. As they all have translational symmetry. A little bit of combinatorial argument along with composition of symmetries can easily reveal why we … Continue reading →

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Existence of a monochromatic rectangle on a lattice containing ‘n’ colors

Posted on August 22, 2020 by Sumant Sumant
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Conway’s paper folding proof of Irrationality of Sqrt(2)

Posted on August 17, 2020 by Sumant Sumant

This is a brilliant proof which requires only paper folding. The idea is to think of the smallest possible integer sides isosceles triangle and one is able to come up with a still smaller integer side triangle and thus contradicting … Continue reading →

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Proof of Pythagorean Theorem Using Similar Triangles

Posted on August 16, 2020 by Sumant Sumant

One of the most common construction in elementary geometry is drawing an altitude in a right angled triangle. This gives rise to 3 Symmetric triangles and one can use this to prove AM-GM inequality. However the same construction can also … Continue reading →

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Geometrical Proof of Irrationality of sqrt(2)

Posted on August 16, 2020 by Sumant Sumant

This is an amazing geometrical proof of the irrationality of . We take a right angled isosceles triangle and assume that all its sides are of integer length. By construction we come up with a smaller isosceles triangle whose sides … Continue reading →

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A clever proof of Pythagorean Theorem

Posted on August 16, 2020 by Sumant Sumant

This is a very elegant proof based on Mnastsakanian radius introduced by Starbird.  

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Mamikon Mnatsakanian’s Idea of Sectors on Polygon

Posted on August 16, 2020 by Sumant Sumant

I came across this on Starbird’s Lecture on Visual Geometry. What Mnatsakanian idea does is to draw sectors of same radius on the extended side of the polygon (It need not be regular). The benefit of this is we get … Continue reading →

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Non periodic Tiling (Pinwheel Tiling)

Posted on August 14, 2020 by Sumant Sumant

Non periodic tiling are fascinating structures. One of the most famous is by Conway and Radin. The following is the base structure for creating a pin wheel tiling. Its a right triangle of side . Unlike a normal triangle which … Continue reading →

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A Fun Fact About Generating Congruent Triangles

Posted on August 14, 2020 by Sumant Sumant

It is quite amazing to know that if we connect the three middle points of a triangle we always obtain for congruent triangles. This fact is used in tessellation. Its really easy to prove that the four triangles are congruent. … Continue reading →

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Sum of Squares of Fibonacci Numbers

Posted on August 13, 2020 by Sumant Sumant

This was an interesting way to introduce Fibonacci number by letting students build  rectangles with blocks of squares and in the process discovering that those blocks need to be Fibonacci numbers. What I really liked about this video is the … Continue reading →

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