Category Archives: geometry

In 4 space dimensions all knots can be unraveled

Nobel laureate Frank Wilczek reminds us in his tutorial article at the Quanta magazine that in 4 space dimensions all knots can be unraveled completely. He explains it as follows: “In three space dimensions, knot theory is a subtle, complicated … Continue reading

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Mobius strip representation of spin 1/2

Physics The elementary particles known as fermions (electron, muon, tau and u, d, c, s, b, t quarks ) and their antiparticles are the constituents of matter. A fermion will impart  units of spin angular momentum when it interacts with … Continue reading

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Klein’s Quartic Curve Inside Out (animation)

This is from 2005 but geometry is ageless. In week 215 John Baez discussed the Klein’s quartic curve. He mentioned the science fiction author and computer programmer Greg Egan’s work on the visualization of the Klein’s quartic curve. I admired … Continue reading

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Cubic curves

The reason I am interested in cubic curves is that they may be the simplest mathematical representations of the twisting action. In “Prometheus and Chronos” I tried to build a conceptual model of particles based on the hypothesis of intrinsic … Continue reading

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Proposals for space-time extensions

Our understanding of the physical universe which is a shadow of the greater Cosmos consisting of spiritual, mental and physical realms is very limited. Our scientific knowledge of the physical universe is growing at an exponential pace but we have only … Continue reading

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image credit I feel so ashamed for not knowing much about al-Biruni – the polymath who lived in the 11’th century. His name was brought to my attention by a newspaper article announcing the establishment of a new private university … Continue reading

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A rotation equals two reflections

In the picture below, the red line and the blue line are the reflection lines. Image credit The A’B’C’ triangle is a reflection of the ABC triangle in the red line. Let’s call this transformation R1 The EDF triangle is … Continue reading

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Quanta Magazine (Simons Foundation)

The Quanta Magazine is a new scientific magazine published online by an editorially independent division of the Simons Foundation. You can reach the magazine at: “Quanta Magazine is an online publication whose mission is to enhance public understanding of … Continue reading

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Internalizing ideas and theories

Apparently the subject of internalization is widely studied in academic psychology and sociology. I am not an expert in the academic aspects of internalization but I am cognizant of my own mental processes. I wanted to share some of my … Continue reading

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Mathematical Spirals

The equations of the basic spirals are given in polar coordinates. The is the radius and is the angle measured from the positive horizontal axis.The and are constants. Archimedes Spiral Fermat Spiral Logarithmic Spiral Golden Spiral The … Continue reading

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Three-Leaved Rose

Simple is beautiful. This universal principle is valid in mathematics and geometry too! Equation for three-leaved-rose (trifolium) in polar coordinates is r = a cos (3θ ) where “a” is a constant and the angle θ is measured from the … Continue reading

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Phase enables relativity

Physicists are experts at assigning new meanings to the terms established in common language. I mentioned the term “dual” or “duality” before. In common language dual means two. In physics dual means equivalent. Physicists use the word “phase” in different … Continue reading

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Platonic Solids Revisited

I will not talk about Sacred Geometry. This is a tutorial on geometry. I guarantee that if you read this tutorial you will learn at least one fun fact that you did not know before. How many 3-dimensional geometric objects … Continue reading

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Dini’s Surface (geometry)

A surface of constant negative curvature obtained by twisting a pseudosphere is known as the Dini’s Surface. It is named after Ulisse Dini. The parametric equations are: x = a cos(u) sin(v) y = a sin(u) sin(v) z = a{cos(v) + … Continue reading

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