Mathematics Learning

LIFE=♾️♾️♾️♾️♾️

A HISTORICAL PHOTO OF NASA SCIENTISTS BEFORE MODERN COMPUTER 🌹❤️♥️

In 1957, long before modern computers, NASA scientists relied on pure mathematical skill, teamwork, and giant chalkboards to solve some of the most complex problems in space exploration. This remarkable photo captures six NASA researchers working together at Systems Labs in California, using ladders to reach every part of an enormous chalkboard filled with equations. Their mission: calculate satellite orbits with accuracy precise enough to withstand the challenges of early spaceflight.
Every formula written on that board represented hours of thought, collaboration, and checking each other’s work by hand. These scientists had to visualize the physics of orbital motion, gravitational pull, angular velocity, and trajectory corrections long before digital tools existed. Mistakes weren’t allowed—every calculation had to be exact, because even a tiny numerical error could send a satellite off course by thousands of miles.
This image is more than a snapshot from the past—it’s a tribute to human intelligence before automated computing took over. It shows how innovation often begins with simple tools, determination, and a shared goal. Today’s space missions use supercomputers and advanced simulations, but the foundation of those systems was built by people who filled giant chalkboards with equations and believed, with absolute focus, that humanity could reach beyond Earth.

AVOGADRO NUMBER 🌹❤️♥️

There is a number that appears in almost every branch of chemistry that connects the invisible world of atoms with the world of jars, scales and grams. It's called Avogadro's number, and it's worth 6.022 × 1023.

To understand why that number exists you have to start from something basic: the atoms are so small that it is impossible to work with them one by one. A single grain of sugar contains more atoms than humans have ever existed in the history of mankind. So how does chemistry do to measure quantities of atoms or molecules practically?.

The answer was to create a counting unit, just like a dozen groups 12 eggs or a thousand groups 1,000 units. In chemistry, that unit is called a mole, and a mole equals exactly 6.022 × 1023 particles, be it atoms, molecules or ions.

The usefulness of that specific number is not arbitrary. A carbon-12 mole weighs exactly 12 grams. A mol of water weighs 18 grams. That correspondence between the number of particles and the mass in grams is what makes the mole so powerful in practice: you can weigh something on a scale and know exactly how many molecules you have.

The name "Avogadro's number" honors the Italian physicist Amedeo Avogadro, who in 1811 formulated the hypothesis that equal volumes of different gases, at the same temperature and pressure, contain the same number of particles. But the one who measured it with experimental accuracy was French physicist Jean Baptiste Perrin, who appears in the picture, at the beginning of the twentieth century studying Brownian motion of particles in suspension. That work earned him the Nobel Prize in Physics in 1926.

Today the Avogadro number is defined with perfect accuracy in the International System of Units: 6.02214076 × 1023. It is one of the seven fundamental values that anchored the entire scientific measurement system in the world.

PETERSEN'S THEOREM 🌹❤️♥️
Petersen's theorem is a foundational result in graph theory that proves every bridgeless cubic graph (a graph where every vertex connects to exactly 3 edges, and no single edge deletion disconnects the graph) contains a perfect matching

1891 (Discovery): The theorem was discovered and published by Danish mathematician Julius Petersen in his pioneering paper on the factorization of regular graphs ("Die Theorie der regulären graphs"). This publication is widely regarded as one of the first explicit and fundamental works in graph theory.

1898 (Counterexample): Petersen presented a famous counterexample graph to challenge a claimed theorem by P.G. Tait regarding the 1-factorability of 3-regular graphs. This counterexample is now celebrated as the Petersen graph. It was the only known "snark" (a connected, bridgeless cubic graph with chromatic index 4) from 1898 until 1946.

1917 & 1926 (Proof Simplifications): Petersen's original proof for the theorem was highly complex and difficult to follow. Mathematicians recognized its deep importance, leading to simplified proofs by O.J. Frink in 1926 (which originally contained a slight flaw).

1936 (Final Proof Correction): Mathematician D. König later corrected the flaw in Frink's proof and incorporated it into the first formal textbook on graph theory, cementing the modern understanding of the theorem.

Today ( June 16) is the 187tt BIRTHDAY anniversary to Julius Petersen.

MATHEMATICS BRIDGEING TODAY TO TOMORROW 🌹❤️♥️⏰⏰⏰⏰👑👑

FRIEDMANN EQUATIONS 🌹❤️♥️

The Friedmann Equations, formulated by Alexander Friedmann, describe the dynamics of an expanding universe in the context of general relativity. These equations relate the rate of expansion, acceleration, gravity, pressure, and the cosmological constant, providing a quantitative framework for cosmology.

Gravity in the Friedmann Equations acts to slow the expansion of the universe, depending on the mass density and pressure of cosmic matter. Conversely, the cosmological constant, often associated with dark energy, contributes a repulsive effect, accelerating the expansion. The interplay between these forces determines the universe’s evolution over time.

The equations predict how the scale factor, representing the relative size of the universe, changes with time. By solving the Friedmann Equations with different mass-energy densities and cosmological constants, scientists can model closed, open, or flat universes, and estimate the age and ultimate fate of the cosmos.

Friedmann’s insights laid the foundation for modern cosmology, including the Big Bang theory and the observation of cosmic expansion. These equations also connect observational astronomy, such as Hubble’s redshift measurements, with theoretical predictions, allowing precise modeling of large-scale cosmic structure.

Culturally, the Friedmann Equations exemplify the power of mathematics in describing the universe. Diagrams showing acceleration, scale factor, and the effects of gravity and dark energy help students and researchers visualize the complex interactions shaping cosmic evolution. Friedmann’s legacy continues to inspire exploration of the universe’s past, present, and future

ROGER PENROSE WHO COMBINED MATH, PHYSICS, GEOMETRY AND COSMOLOGY ❤️🌹♥️

Roger Penrose had a “bumpy ride” at school.

He remembers being forced to nap at his first school in Philadelphia, Pennsylvania but never succeeding in sleeping. In his next school in Ontario, Canada his test scores were lacklustre. Luckily, a teacher named Mr Stenett realised he was a little slow taking tests and could score highly with enough time.

Returning to England after the end of the World War II, he found he was behind with Latin at his new school, but quickly climbed to the top of the class. He impressed his teachers with his natural grasp of mathematics – geometry in particular.

Having initially struggled to work out which area of mathematics to specialise in at university, Penrose obtained his PhD. As well as exploring the Caley form, he took courses of lectures on general relativity and basic quantum mechanics, which would influence his future work. He also forged a strong friendship with Dennis Sciama, who had been a graduate student of the great physicist Paul Dirac with whom he shared an interest in cosmology.

Proving that it can take a little time to achieve academic greatness, Penrose was awarded the Nobel Prize in Physics 2020 “for the discovery that black hole formation is a robust prediction of the general theory of relativity.”

GENERAL RELATIVITY VS PYTHAGORAS 🌹❤️♥️
EXPLANATION OF GENERAL RELATIVITY WITH PYTHAGORAS ONLY🌹❤️♥️

Every week or so, I try to produce one more challenging post. In this one, I want to explain the basic maths behind general relativity using only Pythagoras. Let me warn pedants in advance, that this will not be a rigorous full description (such as is in my book on the subject). However, I hope I can give you a rudimentary understanding of why gravitational time dilation (clocks running slower near a mass such as the earth) results in objects being attracted towards the mass. Everything in this post is based on mainstream physics, as are all my other posts.

For those of you who left high school long ago, Pythagoras' theorem is the following: the square of the length of the hypotenuse of a triangle (the long side) is the sum of the squares of the two other sides. Thus if we label the length of the longest side as "c" and the other two as "a" and "b", then a²+b²=c². That is all we will need.

NEWTON'S WORLD

Let's start in the classical world of Newton. Consider a dung beetle moving through space, as shown in the figure on the right. Why a dung beetle? Well it could be a rocket or a meteorite, but why not a dung beetle - I like them. Anyway, I have labelled the direction of motion as the y axis, and a small change in the beetle's position on this axis as dy. If there is no accelerating force present (ignore the earth for now), then Newton tells us that the beetle's path should be the shortest route that gets from the starting value on the y axis to the finishing value. Clearly this is a straight line along the y axis. This shortest path is called a "geodesic", (typically labelled ds) and its length is dy.

But what if the dung beetle veered in the x direction? Well, the length of the path for the same change dy along the y axis would become, using Pythagoras: ds²=dy²+dx². Clearly this is a longer path. The shortest path is when dx=0 and ds²=dy². Any movement in the x direction increases this. In Newton's classical world the total length ds of small steps in the dx, dy and dz directions is:

ds² = dx² + dy² + dz² Classic Newtonian space

Thus, our friendly dung beetle takes the shortest path and happily whizzes by in a straight line along the y axis. Apologies if this is obvious and boring.

SPECIAL RELATIVITY

Now we are going to spice things up by adding in the fourth dimension: "time". Einstein's starting tenet is that the speed of light is the same for all observers. For this to be true an observer must see a moving clock tick more slowly than the same clock when stationary. The left of the figure explains why.

Let's suppose we view a clock stationary. The clock works by reflecting light to and fro between two mirrors - each "hit" on the mirror is a tick on the clock. This is shown as (a) at the top left of the figure. Let's label the time between each tick as dT (with a capital T), so dT is the length of tick of the stationary clock.

Now let's consider the perspective of an observer who sees the clock moving by. This is shown as (b). I have exaggerated the movement. In fact, as shown, the clock would be moving at half the speed of light! Anyway, for this observer the path of the light between each "hit" of the mirrors is longer, because the clock is moving in the x direction. Let's label the time between each tick of the clock as dt (with a small t) when it is observed moving.

We can superimpose the paths of the light for the clock observed stationary and moving as shown at the bottom of the figure on the left. And we can compare the distances of the paths. Both observers must see the light move at the same speed of light c. That is Einstein's rule. Thus the distance along the stationary path must be c dT (the speed of light multiplied by the time of the tick). Similarly the distance along the moving path must be c dt. Note for the moving path, the clock has moved a distance dx.

This gives a right-angle triangle as shown in the figure and we apply Pythagoras to get: dx² + c²dT² = c²dt². This can be rearranged to: dx² - c²dt² = -c²dT². This can be expanded for motion in all directions:
dx² + dy² + dz² - c²dt² = -c²dT².

Now, this is VERY different from the classical view of Newton. Different observers may see the clock fly by at different speeds. Each speed will have different values for dx, dy, dz and dt (the length of the clock tick), but all will agree on the combination shown above. In summary the key formula for the length ds of a path through spacetime is:

ds² = dx² + dy² + dz² - c²dt² Minkowski spacetime

This quantity is called the invariant interval. The formula gives the distance between two points in spacetime in what is called flat space, no curvature so no gravity, also known as Minkowski spacetime. Note that you must include time in the calculation (the length of tick of a clock) and it appears as a NEGATIVE quantity. This changes everything.

Let me assure sceptics that there is copious evidence for the time dilation related to motion such as sending clocks around the world on jets (the Hafele-Keating experiment) and the increase in half-life of particles moving at high speed (for example muons detected at the earth's surface and particles in the Large Hadron Collider). The list goes on and on.

GENERAL RELATIVITY

This brings us to general relativity. Einstein's idea was that time dilation is not only produced by motion, but also by the presence of mass or energy: this is gravitational time dilation. The tick of a clock is longer in the presence of a gravitational field. We now know this to be true. Our clocks are accurate enough to detect it. For example the clocks on GPS satellites tick slightly faster than those at ground level because, being further away from the centre of the earth, they are in a weaker gravitational field. If we did not adjust for this, GPS systems would drift off by about 10 kilometres per day.

But WHY does gravitational time dilation result in objects moving towards a massive object? Let's repeat the thought experiment that we started with, illustrated on the right of the figure. Let's start with route (1) in the figure. The dung beetle moves dy in a straight line along the y axis. The path length through Minkowski spacetime for route (1) is shown at the bottom of the figure:
ds² = dy² - c²dt².

But what happens if the dung beetle's path veers towards the earth? Let's calculate. Let's assume that in moving dy along the y axis, the dung beetle also moves a small distance dx towards the mass. As a crude model, let's suppose the path is a straight line as shown in route (2) in the figure. As the path takes the dung beetle closer to the earth, there is time dilation, so I use the label dt' to show the length of a clock tick has a new value. This path length through spacetime is shown at the bottom of the figure:
ds² = dy² + dx² - c²dt'²

Gravitational time dilation means that dt' is greater than dt. Moving dx in the x direction adds dx², but the increase in dt' reduces the path length, and the latter is multiplied by the speed of light - the effect of even a tiny change in the tick of a clock is magnified. The direct path along the y axis is NOT the shortest path!

The closer the beetle moves towards the Earth, the greater the time dilation becomes. The gain in the time term therefore becomes larger and larger. The result is not merely a one-off nudge towards the Earth but a continuing tendency for the preferred path through spacetime to bend towards regions of slower time.

I should note that as physicists we tend to talk about maximising proper time rather than minimising pathway distance. However, I think my approach is justified for the simple intuition being developed here.

SUMMARY

Unaccelerated objects follow "geodesics", which can be described in layman's language as the shortest path. In the classical physics of Newton, the shortest path through space between two values along the y axis is a direct line in the y direction.

However, Einstein showed us that time is a fourth dimension. We need to think about paths through spacetime, not space. When you factor in that the speed of light is constant for all observers, the length of a path through spacetime includes time as a negative entry.

Accounting for gravitational time dilation near a mass, you find that the shortest path changes to one that includes moving towards the mass. This is because the resulting time dilation shortens the path length through spacetime. The result is gravitational acceleration towards the mass.

Let me repeat that I am not trying to fully or accurately describe Einstein's theory of general relativity (which I cover in detail in my book on the subject). I just want to give you a taste. If you have worked your way through this, congratulations - you now have a rudimentary understanding.

Quote of Richard Feynman 🌹❤️♥️