3.141592653589793238462643383279502884197169399375105820974944592307816406286208998628034825342117067982148086513282306647093844609550582231725359408128481117450284102701938521105596446229489549303819644288109756659334461284756482337867831652712019091456485669234603486104543266482133936072602491412737245870066063155881748815209209628292540917153643678925903600113305305488204665213841469519415116094330572703657595919530921861173819326117931051185480744623799627495673518857527248912279381830119491298336733624406566430860213949463952247371907021798609437027705392171762931767523846748184676694051320005681271452635608277857713427577896091736371787214684409012249534301465495853710507922796892589235420199561121290219608640344181598136297747713099605187072113499999983729780499510597317328160963185950244594553469083026425223082533446850352619311881710100031378387528865875332083814206171776691473035982534904287554687311595628638823537875937519577818577805321712268066130019278766111959092164201989380952572010654858632788659361533818279682303019520353018529689957736225994138912497217752834791315155748572424541506959508295331168617278558890750983817546374649393192550604009277016711390098488240128583616035637076601047101819429555961989467154357846878861444324224152358623224381745688499164062862089986280348253421170679821480865132823066470938446095505822317253594081284811174502841027019385211055596446229489549303819644288109756659334461284756482337867831652712019091456485669234603486104543266482133936072602491412737245870066063155881748815209209628292540917153643678925903600113305305488204665213841469519415116094330572703657595919530921861173819326117931051185480744623799627495673518857527248912279381830119491298336733624406566430860213949463952247371907021798609437027705392171762931767523846748184676694051320005681271452635608277857713427577896091736371787214684409012249534301465495853710507922796892589235420199561121290219608640344181598136297747713099605187072113499999983729780499510597317328160963185950244594553469083026425223082533446850352619311881710100031378387528865875332083814206171776691473035982534904287554687311595628638823537875937519577818577805321712268066130019278766111959092164201989380952572010654858632788659361533818279682303019520353018529689957736225994138912497217752834791315155748572424541506959508295331168617278558890750983817546374649393192550604009277016711390098488240128583616035637076601047101819429555961989467154357846878861444324224152358623224381745688499164062862089986280348253421170679821480865132823066470938446095505822317253594081284811174502841027019385211055596446229489549303819644288109756659334461284756482337867831652712019091456485669234603486104543266482133936072602491412737245870066063155881748815209209628292540917153643678925903600113305305488204665213841469519415116094330572703657595919530921861173819326117931051185480744623799627495673518857527248912279381830119491298336733624406566430860213949463952247371907021798609437027705392171762931767523846748184676694051320005681271452635608277857713427577896091736371787214684409012249534301465495853710507922796892589235420199561121290219608640344181598136297747713099605187072113499999983729780499510597317328160963185950244594553469083026425223082533446850352619311881710100031378387528865875332083814206171776691473035982534904287554687311595628638823537875937519577818577805321712268066130019278766111959092164201989380952572010654858632788659361533818279682303019520353018529689957736225994138912497217752834791315155748572424541506959508295331168617278558890750983817546374649393192550604009277016711390098488240128583616035637076601047101819429555961989467154357846878861444324224152358623224381745688499164062862089986280348253421170679821480865132823066470938446095505822317253594081284811174502841027019385211055596446229489549303819644288109756659334461284756482337867831652712019091456485669234603486104543266482133936072602491412737245870066063155881748815209209628292540917153643678925903600113305305488204665213841469519415116094330572703657595919530921861173819326117931051185480744623799627495673518857527248912279381830119491298336733624406566430860213949463952247371907021798609437027705392171762931767523846748184676694051320005681271452635608277857713427577896091736371787214684409012249534301465495853710507922796892589235420199561121290219608640344181598136297747713099605187072113499999983729780499510597317328160963185950244594553469083026425223082533446850352619311881710100031378387528865875332083814206171776691473035982534904287554687311595628638823537875937519577818577805321712268066130019278766111959092164201989380952572010654858632788659361533818279682303019520353018529689957736225994138912497217752834791315155748572424541506959508295331168617278558890750983817546374649393192550604009277016711390098488240128583616035637076601047101819429555961989467154357846878861444324224152358623224381745688499164062862089986280348253421170679821480865132823066470938446095505822317253594081284811174502841027019385211055596446229489549303819644288109756659334461284756482337867831652712019091456485669234603486104543266482133936072602491412737245870066063155881748815209209628292540917153643678925903600113305305488204665213841469519415116094330572703657595919530921861173819326117931051185480744623799627495673518857527248912279381830119491298336733624406566430860213949463952247371907021798609437027705392171762931767523846748184676694051320005681271452635608277857713427577896091736371787214684409012249534301465495853710507922796892589235420199561121290219608640344181598136297747713099605187072113499999983729780499510597317328160963185950244594553469083026425223082533446850352619311881710100031378387528865875332083814206171776691473035982534904287554687311595628638823537875937519577818577805321712268066130019278766111959092164201989380952572010654858632788659361533818279682303019520353018529689957736225994138912497217752834791315155748572424541506959508295331168617278558890750983817546374649393192550604009277016711390098488240128583616035637076601047101819429555961989467154357846878861444324224152358623224381745688499164062862089986280348253421170679821480865132823066470938446095505822317253594081284811174502841027019385211055596446229489549303819644288109756659334461284756482337867831652712019091456485669234603486104543266482133936072602491412737245870066063155881748815209209628292540917153643678925903600113305305488204665213841469519415116094330572703657595919530921861173819326117931051185480744623799627495673518857527248912279381830119491298336733624406566430860213949463952247371907021798609437027705392171762931767523846748184676694051320005681271452635608277857713427577896091736371787214684409012249534301465495853710507922796892589235420199561121290219608640344181598136297747713099605187072113499999983729780499510597317328160963185950244594553469083026425223082533446850352619311881710100031378387528865875332083814206171776691473035982534904287554687311595628638823537875937519577818577805321712268066130019278766111959092164201989380952572010654858632788659361533818279682303019520353018529689957736225994138912497217752834791315155748572424541506959508295331168617278558890750983817546374649393192550604009277016711390098488240128583616035637076601047101819429555961989467154357846878861444324224152358623224381745688499164062862089986280348253421170679821480865132823066470938446095505822317253594081284811174502841027019385211055596446229489549303819644288109756659334461284756482337867831652712019091456485669234603486104543266482133936072602491412737245870066063155881748815209209628292540917153643678925903600113305305488204665213841469519415116094330572703657595919530921861173819326117931051185480744623799627495673518857527248912279381830119491298336733624406566430860213949463952247371907021798609437027705392171762931767523846748184676694051320005681271452635608277857713427577896091736371787214684409012249534301465495853710507922796892589235420199561121290219608640344181598136297747713099605187072113499999983729780499510597317328160963185950244594553469083026425223082533446850352619311881710100031378387528865875332083814206171776691473035982534904287554687311595628638823537875937519577818577805321712268066130019278766111959092164201989380952572010654858632788659361533818279682303019520353018529689957736225994138912497217752834791315155748572424541506959508295331168617278558890750983817546374649393192550604009277016711390098488240128583616035637076601047101819429555961989467154357846878861444324224152358623224381745688499164062862089986280348253421170679821480865132823066470938446095505822317253594081284811174502841027019385211055596446229489549303819644288109756659334461284756482337867831652712019091456485669234603486104543266482133936072602491412737245870066063155881748815209209628292540917153643678925903600113305305488204665213841469519415116094330572703657595919530921861173819326117931051185480744623799627495673518857527248912279381830119491298336733624406566430860213949463952247371907021798609437027705392171762931767523846748184676694051320005681271452635608277857713427577896091736371787214684409012249534301465495853710507922796892589235420199561121290219608640344181598136297747713099605187072113499999983729780499510597317328160963185950244594553469083026425223082533446850352619311881710100031378387528865875332083814206171776691473035982534904287554687311595628638823537875937519577818577805321712268066130019278766111959092164201989380952572010654858632788659361533818279682303019520353018529689957736225994138912497217752834791315155748572424541506959508295331168617278558890750983817546374649393192550604009277016711390098488240128583616035637076601047101819429555961989467154357846878861444324224152358623224381745688499164062862089986280348253421170679821480865132823066470938446095505822317253594081284811174502841027019385211055596446229489549303819644288109756659334461284756482337867831652712019091456485669234603486104543266482133936072602491412737245870066063155881748815209209628292540917153643678925903600113305305488204665213841469519415116094330572703657595919530921861173819326117931051185480744623799627495673518857527248912279381830119491298336733624406566430860213949463952247371907021798609437027705392171762931767523846748184676694051320005681271452635608277857713427577896091736371787214684409012249534301465495853710507922796892589235420199561121290219608640344181598136297747713099605187072113499999983729780499510597317328160963185950244594553469083026425223082533446850352619311881710100031378387528865875332083814206171776691473035982534904287554687311595628638823537875937519577818577805321712268066130019278766111959092164201989380952572010654858632788659361533818279682303019520353018529689957736225994138912497217752834791315155748572424541506959508295331168617278558890750983817546374649393192550604009277016711390098488240128583616035637076601047101819429555961989467154357846878861444324224152358623224381745688499164062862089986280348253421170679821480865132823066470938446095505822317253594081284811174502841027019385211055596446229489549303819644288109756659334461284756482337867831652712019091456485669234603486104543266482133936072602491412737245870066063155881748815209209628292540917153643678925903600113305305488204665213841469519415116094330572703657595919530921861173819326117931051185480744623799627495673518857527248912279381830119491298336733624406566430860213949463952247371907021798609437027705392171762931767523846748184676694051320005681271452635608277857713427577896091736371787214684409012249534301465495853710507922796892589235420199561121290219608640344181598136297747713099605187072113499999983729780499510597317328160963185950244594553469083026425223082533446850352619311881710100031378387528865875332083814206171776691473035982534904287554687311595628638823537875937519577818577805321712268066130019278766111959092164201989380952572010654858632788659361533818279682303019520353018529689957736225994138912497217752834791315155748572424541506959508295331168617278558890750983817546374649393192550604009277016711390098488240128583616035637076601047101819429555961989467154357846878861444324224152358623224381745688499164062862089986280348253421170679821480865132823066470938446095505822317253594081284811174502841027019385211055596446229489549303819644288109756659334461284756482337867831652712019091456485669234603486104543266482133936072602491412737245870066063155881748815209209628292540917153643678925903600113305305488204665213841469519415116094330572703657595919530921861173819326117931051185480744623799627495673518857527248912279381830119491298336733624406566430860213949463952247371907021798609437027705392171762931767523846748184676694051320005681271452635608277857713427577896091736371787214684409012249534301465495853710507922796892589235420199561121290219608640344181598136297747713099605187072113499999983729780499510597317328160963185950244594553469083026425223082533446850352619311881710100031378387528865875332083814206171776691473035982534904287554687311595628638823537875937519577818577805321712268066130019278766111959092164201989380952572010654858632788659361533818279682303019520353018529689957736225994138912497217752834791315155748572424541506959508295331168617278558890750983817546374649393192550604009277016711390098488240128583616035637076601047101819429555961989467154357846878861444324224152358623224381745688499164062862089986280348253421170679821480865132823066470938446095505822317253594081284811174502841027019385211055596446229489549303819644288109756659334461284756482337867831652712019091456485669234603486104543266482133936072602491412737245870066063155881748815209209628292540917153643678925903600113305305488204665213841469519415116094330572703657595919530921861173819326117931051185480744623799627495673518857527248912279381830119491298336733624406566430860213949463952247371907021798609437027705392171762931767523846748184676694051320005681271452635608277857713427577896091736371787214684409012249534301465495853710507922796892589235420199561121290219608640344181598136297747713099605187072113499999983729780499510597317328160963185950244594553469083026425223082533446850352619311881710100031378387528865875332083814206171776691473035982534904287554687311595628638823537875937519577818577805321712268066130019278766111959092164201989380952572010654858632788659361533818279682303019520353018529689957736225994138912497217752834791315155748572424541506959508295331168617278558890750983817546374649393192550604009277016711390098488240128583616035637076601047101819429555961989467154357846878861444324224152358623224381745688499164062862089986280348253421170679821480865132823066470938446095505822317253594081284811174502841027019385211055596446229489549303819644288109756659334461284756482337867831652712019091456485669234603486104543266482133936072602491412737245870066063155881748815209209628292540917153643678925903600113305305488204665213841469519415116094330572703657595919530921861173819326117931051185480744623799627495673518857527248912279381830119491298336733624406566430860213949463952247371907021798609437027705392171762931767523846748184676694051320005681271452635608277857713427577896091736371787214684409012249534301465495853710507922796892589235420199561121290219608640344181598136297747713099605187072113499999983729780499510597317328160963185950244594553469083026425223082533446850352619311881710100031378387528865875332083814206171776691473035982534904287554687311595628638823537875937519577818577805321712268066130019278766111959092164201989380952572010654858632788659361533818279682303019520353018529689957736225994138912497217752834791315155748572424541506959508295331168617278558890750983817546374649393192550604009277016711390098488240128583616035637076601047101819429555961989467154357846878861444324224152358623224381745688499164062862089986280348253421170679821480865132823066470938446095505822317253594081284811174502841027019385211055596446229489549303819644288109756659334461284756482337867831652712019091456485669234603486104543266482133936072602491412737245870066063155881748815209209628292540917153643678925903600113305305488204665213841469519415116094330572703657595919530921861173819326117931051185480744623799627495673518857527248912279381830119491298336733624406566430860213949463952247371907021798609437027705392171762931767523846748184676694051320005681271452635608277857713427577896091736371787214684409012249534301465495853710507922796892589235420199561121290219608640344181598136297747713099605187072113499999983729780499510597317328160963185950244594553469083026425223082533446850352619311881710100031378387528865875332083814206171776691473035982534904287554687311595628638823537875937519577818577805321712268066130019278766111959092164201989380952572010654858632788659361533818279682303019520353018529689957736225994138912497217752834791315155748572424541506959508295331168617278558890750983817546374649393192550604009277016711390098488240128583616035637076601047101819429555961989467154357846878861444324224152358623224381745688499164062862089986280348253421170679821480865132823066470938446095505822317253594081284811174502841027019385211055596446229489549303819644288109756659334461284756482337867831652712

What was the last lecture you attended? ✍️

Paradoxes in Physics and Mathematics ✍️ 1. Zeno's Paradoxes: A set of philosophical problems that challenge the concept of motion and continuity. For example, Achilles and the tortoise paradox argues that Achilles can never overtake a tortoise given a head start, as he must first reach the point where the tortoise began. 2. Russell's Paradox: A contradiction found in naive set theory, where the set of all sets that do not contain themselves leads to a logical inconsistency. This paradox highlights problems in defining a universal set. 3. The Barber Paradox: A self-referential paradox where a barber shaves all those who do not shave themselves. The question arises: does the barber shave himself? If he does, he shouldn't; if he doesn't, he should. 4. The Liar Paradox: A statement that declares itself false, such as "This statement is false." If it's true, then it must be false, and vice versa, leading to a contradiction. 5. The Sorites Paradox: This paradox arises from vague predicates, such as "heap." If removing a single grain of sand from a heap doesn’t stop it from being a heap, how many grains can be removed before it ceases to be a heap? 6. The Monty Hall Problem: A probability puzzle based on a game show scenario. Given a choice among three doors (with a prize behind one), switching after one non-prize door is revealed increases your chances of winning from 1/3 to 2/3. 7. The Birthday Paradox: Refers to the surprising probability that in a group of just 23 people, there's about a 50% chance that at least two individuals share the same birthday, challenging intuitive understanding of probability. 8. The Banach-Tarski Paradox: A theorem in set-theoretic geometry that states a solid ball in 3-dimensional space can be split into a finite number of pieces that can be reassembled into two identical copies of the original ball. This paradox challenges our understanding of volume and measure. 9. The Twin Paradox: A thought experiment in relativity where one twin travels at a significant fraction of the speed of light while the other remains on Earth. Upon reunion, the traveling twin is younger than the stationary twin, illustrating the effects of time dilation. 10. The Grandfather Paradox: A time travel paradox where a person travels back in time and inadvertently prevents their grandfather from meeting their grandmother, thereby preventing their own existence. This raises questions about causality and the nature of time.

The position probability distributions of eight stationary states of hydrogen atom: They show the likelihood of finding the electron in a given region of space around the nucleus, depending on the quantum numbers n, l, and m. The wave functions are solutions to the Schrödinger equation for the hydrogen atom, and they depend on three variables: r, θ, and φ, which are the spherical coordinates of the electron. The eight stationary states shown in the image are: The ground state (n = 1, l = 0, m = 0), which has a spherical shape and is symmetric around the nucleus. The probability density is highest at the nucleus and decreases exponentially with increasing r. The first excited state (n = 2, l = 0, m = 0), which has a spherical shape and is symmetric around the nucleus. The probability density has a node at r = 2a0, where a0 is the Bohr radius, and peaks at r = a0 and r = 4a0. The second excited state (n = 2, l = 1, m = 0), which has a dumbbell shape and is symmetric along the z-axis. The probability density has a node at the nucleus and peaks at r = a0 for θ = π/2. The third excited state (n = 2, l = 1, m = ±1), which has a dumbbell shape and is symmetric along the x-axis or y-axis, depending on the sign of m. The probability density has a node at the nucleus and peaks at r = a0 for θ = π/4 or θ = 3π/4. The fourth excited state (n = 3, l = 0, m = 0), which has a spherical shape and is symmetric around the nucleus. The probability density has two nodes at r ≈ 1.6a0 and r ≈ 4.8a0, and peaks at r ≈ a0, r ≈ 3a0, and r ≈ 6a0. The fifth excited state (n = 3, l = 1, m = 0), which has a dumbbell shape and is symmetric along the z-axis. The probability density has two nodes at r ≈ 2.5a0 and r ≈ 5a0 for θ = π/2, and peaks at r ≈ a0 and r ≈ 4a0 for θ = π/2. The sixth excited state (n = 3, l = 1, m = ±1), which has a dumbbell shape and is symmetric along the x-axis or y-axis, depending on the sign of m. The probability density has two nodes at r ≈ 2.5a0 and r ≈ 5a0 for θ = π/4 or θ = 3π/4, and peaks at r ≈ a0 and r ≈ 4a0 for θ = π/4 or θ = 3π/4. The seventh excited state (n = 3, l = 2, m = 0), which has a cloverleaf shape and is symmetric along the z-axis. The probability density has three nodes: one at the nucleus, one at r ≈ a0 for θ ≈ π/6 or θ ≈ 5π/6, and one at r ≈ a0 for θ ≈ π/2. It peaks at r ≈ a0 for θ ≈ π/3 or θ ≈ 2π/3. The image also shows how the position probability distributions are projected along the y-axis by integrating over x and z. This gives an idea of how the electron cloud appears when viewed from above or below. 📷 created with David Manthey's free Orbital Viewer by Ulrich Mohrhoff

The 7 classical indeterminate forms: ✍️ 1. 0/0 2. ∞/∞ 3. 0 × ∞ 4. ∞ − ∞ 5. 0⁰ 6. ∞⁰ 7. 1^∞

[L-R] Two titans, Niels Bohr and J. Robert Oppenheimer conversing, 1950. Credit : Niels Bohr Archive.

Albert Einstein's grade sheet, with 5s and 6s(Out of 6) in Physics, and 4s in most of the Math courses.

J. Robert Oppenheimer’s hat is seen resting on cyclotron pipes, 1946. Credit : Los Alamos National Laboratory

What’s a scientific concept you once found challenging but now find fascinating? ✍️

One of the earliest examples of a systematic approach to empirical inquiry is found in the Edwin Smith papyrus, an ancient Egyptian medical textbook from c. 1600 BCE, which describes the examination, diagnosis, treatment, and prognosis of various diseases.

Thank this man for digital communication, data compression (ZIP, MP3), coding theory, and cryptography. Claude Shannon’s biggest scientific contribution was the creation of information theory in his 1948 paper “A Mathematical Theory of Communication.” He defined “information” mathematically, introducing the concept of bits (binary digits) as the fundamental unit. He borrowed the term from thermodynamics to describe the uncertainty or information content in a message. Shannon entropy became the foundation of data compression and coding. He proved the Shannon limit (channel capacity theorem): there is a maximum rate (capacity) at which data can be transmitted over a noisy channel with arbitrarily low error, using proper encoding. His framework underlies digital communication, data compression (ZIP, MP3), coding theory, cryptography, the internet, mobile phones, and AI.

The holographic principle is a theory in physics suggesting that all the information contained within a volume of space can be represented on the boundary of that space. In essence, our 3D universe might be like a hologram, with the "real" information encoded on a 2D surface at the edges of the universe. This idea emerged from black hole physics, where the information about objects falling into a black hole seems to be stored on its 2D event horizon. It has since been extended to the universe as a whole, implying that our perception of three dimensions might be a projection of deeper, 2D laws of physics.

The first edition of Newton's Philosophiæ Naturalis Principia Mathematica, financed by Edmond Halley for the Royal Society, was published on 5 July 1687.

On this day 14 years ago, the discovery of Higgs Boson was announced by CERN.

We, by our arts may be called the grandsons of God. -- Leonardo da Vinci

Max Planck was a musical prodigy who could play the piano, organ, and cello. He even composed his own songs and operas. He once said, “The main source of all the greatest achievements in natural science, I am convinced, lies in the divine gift of musicality.”

Recommend a science book that everyone should read ✍️

God created everything by number, weight and measure. - Sir Isaac Newton

1953: Alan Turing’s handwritten letter to a friend where he explains his Solitaire method and advice on how to play the game.

"God does not play dice with the universe." - A. Einstein

Why do the laws of physics fail at Singularity? ✍️ The laws of physics break down at singularities because the extreme conditions there exceed the limits of our current understanding of the universe. Singularities, such as the center of a black hole or the moment of the Big Bang, are points where density becomes infinite, and spacetime curvature becomes immeasurable. Here’s why they defy physics: 1. General Relativity Fails: Einstein’s equations can’t handle infinite values, rendering them ineffective in such conditions. 2. Quantum Mechanics Takes Over: At very small scales, quantum effects dominate, but we lack a complete theory of quantum gravity to unify it with relativity. 3. Infinite Spacetime Distortion: The extreme curvature of spacetime at singularities breaks conventional mathematical models. 4. Observational Limits: Singularities are often hidden by event horizons (like in black holes), making direct study impossible.