Category Archives: mathematics

A profile of Roger Penrose by Philip Ball

I have mentioned several times in this blog that Roger Penrose is one of my three heroes in physics – others being Albert Einstein and Gerard ‘t Hooft. The UK magazine Prospect published a profile of Roger Penrose. I enjoyed this … Continue reading

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Geometrical versus Topological

NYC geographical map shown below is an example of geometrical representation. The distances between the boroughs of NYC are proportional to the actual distances. NYC subway diagram shown below is an example of topological representation. On this topological map the distances between the boroughs … Continue reading

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Atiyah lecture at 2016 Heidelberg Laureate Forum

Image credit Before you watch Atiyah’s wonderful lecture given at the Heidelberg Laureate Forum you should first read these two blogpost by Ben Orlin: 21 Essential Quotes from Sir Michael Atiyah interview (Quanta Magazine): Michael Atiyah’s Imaginative State of Mind … Continue reading

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LMFDB: unifying the building blocks of mathematics

image credit: http://www.lmfdb.org The LMFDB is a new web resource developed by a large team of mathematicians around the world. The www.lmfdb.org describes LMFDB as the database of L-functions, modular forms, and related objects. L-functions are like the DNA of mathematics. The article … Continue reading

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Centennial Anniversary of Einstein’s General Relativity Theory

100 years a ago on November 25 1915 Albert Einstein presented the final version of his theory of gravitation known as the General Relativity to Prussian Academy of Sciences. The paper was published in 1916. Why do we mention David Hilbert in the … Continue reading

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Polchinski’s review paper: Dualities of Fields and Strings

I have read Joseph Polchinski’s excellent review paper titled “Dualities of Fields and Strings“. This paper is written for experts. I am not an expert in fields and strings. I am not an expert in dualities either but I was able … 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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Al-Biruni

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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Maryam Mirzakhani: the first woman to win a Fields Medal

Image credit Maryam Mirzakhani — a 37-year-old mathematics professor at Stanford University — was awarded a Fields Medal. She is the first woman to win a Fields Medal – the most prestigious prize in mathematics. The Fields Medal is awarded every four … Continue reading

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Credence

I saw the word “credence” in a blog post by Sean Carroll.  He defines credence as “degree of belief.” Here’s an interesting quote from that blog post “Quantum probabilities are really credences — statements about the best degree of belief … Continue reading

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A selection of resources on mathematics, physics and philosophy

In my previous post I mentioned the existence of a “collective mechanism” that pays attention to improvements/innovations in individual expressions. I try to do my part and bring the reader’s attention to excellent educational resources on math, physics and philosophy. I … Continue reading

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My Response to Hut-Alford-Tegmark

This is the 3rd installment of a 3 part series. The first two were: Piet Hut the Hero Physicist Hut-Alford-Tegmark Debate on Math, Matter and Mind In this piece I will comment on the Hut-Alford-Tegmark debate. For their debate Hut-Alford-Tegmark used a … Continue reading

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Hut-Alford-Tegmark Debate on Math, Matter and Mind

This is a follow-up article on “Piet Hut the Hero Physicist” where I mentioned Hut’s interesting paper titled “On Math, Matter and Mind” [2]. I will provide an introduction to his paper here. You can read my comments on that … Continue reading

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Angle

An angle is always defined by 2 numbers Typically those 2 numbers are related to 2 line segments that have a common end point. The coordinate system or the nature of the space in which these line segments are embedded … Continue reading

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π Day

March 14 is the day. Coincidentally, March 14 is also the birthday of Albert Einstein. The best encyclopedic resource for anything related to mathematics is Wolfram MathWorld which is created, developed and nurtured by Eric Weisstein. I have tremendous respect … Continue reading

Posted in mathematics, society, tutorial

Atiyah’s Grand Survey of Duality in Mathematics and Physics

Michael Francis Atiyah, OM, FRS, FRSE, FAA is a British mathematician specializing in geometry. He was awarded the Fields Medal in 1966, the Copley Medal in 1988, and the Abel Prize in 2004. Michael Atiyah was a professor at both … Continue reading

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Einstein and Gödel

When I was younger I was deeply impressed by Douglas R. Hofstadter’s book “Gödel, Escher, Bach: An Eternal Golden Braid.” The “Gödel” in the title of that book refers to Kurt Gödel who was one of the greatest mathematicians of … 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:  https://www.simonsfoundation.org/quanta/ “Quanta Magazine is an online publication whose mission is to enhance public understanding of … Continue reading

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Three Worlds and Three Mysteries of Penrose

“Platonic-mathematical, physical, and mental – has its own kind of reality, and where each is (deeply and mysteriously) founded in the one that preceeds it (the worlds being taken cyclicly). I like to think that , in a sense, the … Continue reading

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Few Comments on the Pair Concept

In my previous post I mentioned that Complex number can be thought of as a pair of real numbers. Quaternion can be thought of as a pair of complex numbers Octonion can be thought of as a pair of quaternions … Continue reading

Posted in mathematics, physics

Cayley–Dickson Construction of Complex, Quaternion and Octonion Numbers

Complex number can be thought of as a pair of real numbers. Quaternion can be thought of as a pair of complex numbers Octonion can be thought of as a pair of quaternions When you construct the complex, quaternion and … Continue reading

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Jacob Barnett the young physicist

I sometimes write about a hypothetical “young physicist” who would change the current paradigm in theoretical particle physics. In All Particles Are God Particles I wrote: Someday, a young physicist with the right background will get inspired and work out … 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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Geometric Algebra (1)

Geometric algebra is not to be confused with algebraic geometry. Geometric algebra is also known as Clifford algebra which has many applications in physics and engineering. Algebraic geometry, on the other hand, is a branch of abstract mathematics. Geometric algebra … Continue reading

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Geometric interpretation of sqrt(-1)

I wrote a post titled Euler rule is almost mystical where I mentioned the Cotes-Euler identity Euler rule is a special case of the Cotes-Euler identity. represents the rotations of the complex unit vector. This provides a natural interpretation for … Continue reading

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Biquaternion

In the previous post we have seen that a quaternion is defined as where and , , , are real numbers. Biquaternion The biquaternion is a complexified quaternion where , , , are complex numbers. The basis elements , , … Continue reading

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Quaternion

Quaternions are 4-dimensional generalizations of complex numbers. It can also be said that the basis elements , , of a quaternion are 3-dimensional generalization of the pure imaginary number . William Rowan Hamilton discovered quaternions in 1843. After many years … 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 http://mathworld.wolfram.com/ArchimedesSpiral.html Fermat Spiral http://mathworld.wolfram.com/FermatsSpiral.html Logarithmic Spiral http://mathworld.wolfram.com/LogarithmicSpiral.html Golden Spiral The … Continue reading

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Golden Number and Lucas Sequence

The golden number is . There is extensive literature on the golden number which is also known as the golden ratio. The exposition in Wolfram Mathworld [http://mathworld.wolfram.com/GoldenRatio.html] is excellent. John Baez gave an interesting exposition [http://math.ucr.edu/home/baez/week203.html]. Mario Livio’s book “The Golden … 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

Posted in geometry, mathematics, physics, science, tutorial

Euler rule is almost mystical

In his wonderful book “Road to Reality” Roger Penrose mentions that the Euler rule is “almost mystical” because it relates the 5 fundamental numbers 0, 1, i, π and e to each other. The Euler rule is a special case … Continue reading

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i raised to the power of i

Remember the general rule Using this rule we can compute i raised to the power of i remember that and

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Zero

Symbols for zero appeared in many cultures Before Common Era (BCE). The philosophical conception of zero, however, is usually attributed to the Indian philosophy. The evidence is in the Sanskrit language. The Sanskrit word “shunya” meaning “emptiness” or “void” was … Continue reading

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Super Modeler Function

Data modeling capabilities of this function is limitless. X is a continous variable. S, R, T are constants. By changing the constants see how the Super Modeler Function is behaving. In the graphs below the horizontal axis is the X … Continue reading

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