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Do two photons approach each other at twice the speed of light, 2 c ? This popular question highlights the distinction between relative and closing speeds in Newtonian physics and their interpretation within the context of special relativity. In everyday Newtonian usage, when we mention that two cars are approaching each other at, say, 100 km/h, we typically refer to the closing speed, determined as the sum of the speeds of both cars. However, the relative speed is also calculated by adding the two speeds. Therefore, in Newtonian terms, the result is the same, so we don't need to clarify which speed we mean. This is not the case in special relativity, where closing and relative speeds are handled quite differently. Specifically, the closing speed of approaching photons is determined as 2c, whereas their relative speed is considered undefined. Fig. 1 . In special relativity, when two photons approach each other head-on, their closing speed is determined as 2c, whereas their relative speed is considered undefined. Before exploring the reasons behind these nuances and their implications in the realm of massless particles, let's clarify the difference between the relative and closing speeds in the case when two cars approach each other head-on: Relative speed  is the speed of one car as measured by an observer inside the second car. The frame of reference is aligned with the second car. Closing speed  is the speed at which the distance between two cars is closing. It is calculated by adding the speeds of these cars as they are measured by an observer on the ground. The frame of reference is aligned with the ground. We begin with Newtonian mechanics. Consider two cars, one red and one blue, approaching each other head-on at speeds V1 and V2 ( Fig. 2 ). An observer on the ground using a radar gun will measure their individual speeds as V1 and V2 , just as they appear on the cars' speedometers. The closing speed , Vc = (V1 + V2) , represents the rate at which the gap between these cars is closing. It is mainly used to calculate the time and location of the cars' meeting point. For instance, if the cars start their journey with a distance D between them, they will meet in time T = D ÷ (V1 + V2) . Knowing their travel time T , we can compute the point at which they converge. It is important to realize that when we add V1  and V2  to determine the closing speed, nothing is actually traveling at this combined speed. The gap is not moving; it simply shrinks. The distance between the cars decreases because each car reduces it from its side at its own rate. Fig. 2 . The closing speed, Vc = ( V1 + V2 ), allows for calculating the travel time T and the collision point of the red and blue cars. Let's place an observer with the radar gun inside a blue car. Since the observer is now stationary relative to this car, he will perceive the red car's speed as a relative speed Vrel = (V1 + V2) . His radar gun will also measure the red car's speed as Vrel = (V1 + V2) , because the radar signal must travel to that car and back to complete the task, effectively measuring the rate at which the gap between them is closing. This is why, in Newtonian physics, the distinction between closing speed and relative speed can become blurred. Shifting the frame of reference from the ground to one of the cars provides an additional viewpoint to help us evaluate the situation from different angles and reach an objective conclusion. Einstein's relativity was born out of the need to reconcile Newtonian physics with the discovery that the speed of light, c, is a universal constant. In this sense, relativity is not a new law, but rather a reinterpretation of existing laws viewed from the standpoint of the new constant. In Newtonian relativity, velocities add. Therefore, if an observer on the ground measures the speeds of a rocket and an approaching photon as 0.1c and c, respectively, then an observer in the rocket should have measured the photon's speed as 1.1c. However, this was not the case. Experimental data demonstrate that the speed of light remains the constant c, unaffected by the position of the observer. Guided by the constancy of c and the uniformity of natural laws across space, time, and frames, Einstein modified Newtonian equations to incorporate the new advancements in physics. This led to the development of special relativity, which is founded on two postulates: The laws of physics remain the same in all inertial frames The speed of light in a vacuum remains constant for all observers, irrespective of their                 or the light source motion . Einstein had no issue with adding up the photon speeds ( c + c ) in the closing speed because it did not conflict with the constancy of c. The closing speed is determined by summing the speeds of the particles measured from the ground. In the lab (ground) frame, the speed of each photon is measured as c. Though we add these speeds together ( Fig. 3 ), nothing actually travels at the combined speed of 2c, just as nothing traveled at the combined speed (V1 + V2) in the example with two cars in Fig. 2 . The speed addition here is merely a mathematical operation, utilising the fact that two particles with equal speeds cover equal distances at the same time, meeting at the midpoint of their journey. No frame transition takes place. Consider two photons in Fig. 3 , one red and one blue, each travelling at a speed of c = 300,000 km/s ( 186,000 miles per second ) . The distance between the photons shrinks at a rate of 2c = 600,000 km/sec, because each photon shortens it by 300,000 km every second from its side. We add their speeds (c + c) to   determine   when and where the photons will meet. However, this addition represents their concurrent motion, not the doubled speed. Nothing is moving at 2c = 600,000 km/sec. We merely have two photons, each independently covering 300,000 km ( 186,000 miles) every second. Their concurrent motion means that they double the mileage, not the speeds. Fig. 3 . The closing speed between two photons is computed as 2c = c + c. However, nothing is moving at this speed. We merely have two photons, each traveling at c and collectively covering twice the distance every second. A different approach to relative speed in special relativity arises from the constraints imposed by the constancy of c. While Einstein never specifically discussed the case of two photons traveling toward each other, he conducted a thought experiment in which he envisioned himself running alongside the light beam. He realized that being stationary relative to the beam ( Fig. 4 ) would result in observing a frozen electromagnetic wave, contradicting Maxwell's equations. This insight led to the conclusion that the observer can't “ride a photon” and ask how fast another photon approaches, because a photon doesn’t have a valid rest frame. In a valid frame, such as one aligned with a laboratory on Earth, the observer would see each photon moving at the speed of c. T he question of how one photon perceives another remained unanswered. Fig. 4 . The observer placed on the electromagnetic wave will move with it at the speed of light, c., perceiving the wave stationary relative to him. Einstein's formal stance was that the relative speed between two photons can't be determined in their own frames, as we can't assign a zero value to the speed of light, c. To me, this formalism seems an evasion using scientific terminology. Frames of reference were specifically designed for conducting thought experiments, such as the one Einstein conducted himself. They are a valuable tool in scientific study that allows a researcher to gain an additional viewpoint. When Einstein aligned his perspective with an electromagnetic wave, he gained an insight that had never been considered before, leading to the development of the theory of relativity. This breakthrough in science came from assigning a reference frame to a photon. So, do we really need to abandon this conceptual tool that has served us so well for centuries just because we got a little spooked by what we saw? Placing an observer on a photon doesn't mean we've stopped the light, just as positioning an observer on the ground doesn't imply that we've stopped the Earth. It merely indicates that the observer has adopted the photon's speed and is stationary relative to it. In classical physics, the relative speed between two bodies is measured from the reference frame of one of them. We will expand this definition into the realm of massless particles by positioning the observer on a photon and exploring the impact of the constancy of c on his measurements. The universe has no special rules for massless particles, as the laws of physics must remain consistent across space, time, and frames for all its constituents. Drawing parallels with Newtonian perspectives will help us develop an intuitive understanding of Einstein's relativity and make it accessible to everyone. Essentially, we will attempt to "ride a photon" and see where it takes us. In the car example, we used a radar gun to measure the relative speed between the vehicles. In our thought experiment, we will also place an observer on one of the photons, say the blue one, and have him measure the speed of the red photon using the radar gun ( Fig.5 ). The radar gun operates by emitting light signals. When two objects are in relative motion, light waves undergo a frequency shift due to the Doppler effect. When the gun positioned on the first object emits a light signal, the signal travels to the other object, bounces off it, and returns to the emitter. The difference in the emitted and received frequencies determines the relative speed between the objects. If there is no relative motion, the radar gun detects zero Doppler effect. Fig. 5 . The observer on the blue photon attempts to measure the relative speed between two photons with the radar gun.. Einstein would be unable to observe a frozen electromagnetic wave because light can only be detected when it interacts with an eye or a measuring device. Therefore, our thought experiment begins with the observer sending a radar signal to the blue photon to ensure there is no relative motion between them, and the observer is aligned with the photon's frame. The observer then sends a radar signal toward the red photon. Since the speed of light is independent of the motion of the light emitter, the signal will travel at speed c alongside the gun and will only reach the red photon at the collision point. Consequently, no signal will be exchanged between the photons during their journey, meaning no relative speed can be measured using the Doppler effect or any other method. In a Newtonian framework, we can determine relative speed because we can use signals that travel significantly faster than cars, allowing for continuous exchange of signals between the vehicles. If we had signals traveling faster than light, we could also determine the photons' relative speed and express its value by using Lorentz transformations, where the speed of light, c , would be replaced with the new, higher constant. However, since no such faster signal exists, the information between the photons can't be exchanged faster than their speed, c . As a result, one photon can receive the information about the other only at their meeting point. We have concluded that the closing speed of two approaching photons is 2 c . This finding does not contradict the postulates of special relativity, as nothing is actually traveling at that speed. Each photon travels at the speed of light, c , as observed in the lab frame. The addition ( c  + c  = 2 c ) is merely used to reflect the fact that the photons cover double the distance due to their concurrent motion. This information is utilized when calculating their travel time and meeting point. In our thought experiment, we attempted to measure the relative speed of two photons, similar to how we measure the relative speed of two head-on cars in Newtonian relativity. We placed an observer with a radar gun on one photon to gauge the speed of the second photon. We found that the relationship between photons can't be described using the classical concept of relative velocity, because the speed of information can't exceed the speed of light c . Since no data can be exchanged between the photons, the notion of how one object perceives the speed of the other object becomes meaningless. By assigning a reference frame to the photon, we have arrived at the same conclusion as Einstein’s special relativity, but through a more intuitive approach, drawing parallels with the familiar Newtonian world. We determined that observers in all frames, including both the lab and photon frames, will measure the speed of light as c . The observer in the photon's frame identifies its own speed as c by comparing it to the speed of the light signal emitted by the radar gun.

We’ve all been there. You're driving down a long, deserted country road with no rest stop in sight. Sooner or later, nature calls — and you pull over, duck behind some bushes, and hope to finish before anyone passes by. The last thing you expect when leaving the scene is for the evidence to follow you, revealing your secrets to the world. Yet, that is exactly what happens in space. A spacecraft can't stop to dispose of waste, so anything that exits the craft will follow it until the difference in velocities eventually separates them. Fig. 1 . An artistic depiction of a spacecraft encircled by frozen droplets. When nature calls in orbit. In the early days of space flight, there were no fancy recycling systems or closed-loop plumbing on board. Astronauts aboard missions like Gemini and Apollo had a simple, if inelegant, solution: they vented urine directly into space. As soon as the liquid left the spacecraft, it met the vacuum of space and instantly froze. Millions of tiny ice crystals formed, spreading out in a glittering cloud that drifted alongside the ship. In sunlight, the frozen droplets caught and reflected light in all directions, creating a shimmering halo that looked, for all the world, like a miniature galaxy. It was, quite literally, the most beautiful bathroom break in history.  The "Constellation Urion". Astronaut Wally Schirra, ever the joker, once called it "Constellation Urion" after seeing one of these sparkling displays during the Gemini missions. And Apollo 9 astronaut Rusty Schweickart described it later as "one of the most beautiful sights in orbit... a spray of sparklers at sunset. "The sight wasn't just funny, it was genuinely mesmerising. During an Apollo urine dump, the crew would time the venting near orbital sunrise or sunset, when sunlight hit the frozen cloid at just the right angle. For a few minutes, the ship would be surrounded by what looked like a swarm of diamonds. Fig. 2 . NASA astronauts: Wally Schirra (left) and Rusty Schweickart (right) Science in the sparkle. Behind the poetry was some practical physics. In orbital microgravity , liquids are weightless. They don't fall; they flow. When exposed to the vacuum of space, water (and, yes, urine) rapidly boils away and freezes simultaneously, forming a trail of fine, reflective ice crystals. These particles gradually disperse, but in the meantime, they can hang around the spacecraft, even reflecting sunlight toward the windows. Mission controllers, meanwhile, had to keep track of these ventings, not for hygiene, but because the escaping mass could subtly alter the spacecraft’s trajectory. Even bodily fluids could knock a billion-dollar rocket off course, if only by a fraction. From dump to drink. Fast-forward to today, and things have come considerably more refined. On the International Space Station, nothing is wasted. Modern life-support systems can reclaim and purify urine into clean drinking water, a process astronauts wryly summarize as: "Yesterday's coffee becomes tomorrow's coffee." The sparkling clouds are gone, but their legend remains – a small, glittering reminder of how space travel turns even the most mundane human needs into something extraordinary. When astronomers mistook pee for a UFO. On more than one occasion, skywatchers on Earth have spotted these glittering plumes from orbiting spacecraft and mistaken them for something mysterious: a strange glow, a comet fragment, or even a visiting spaceship from beyond the stars. The sight set off a flurry of excited reports and speculation among amateur astronomers. Was it space debris? A meteor shower? A new phenomenon in the upper atmosphere? Little did they know, the glowing visitor they admired wasn't an alien at all. Its origins were much more mundane and down-to-earth. A glittering cloud silently twinkling across the night sky was not a UFO but an astronaut's urine frozen in the near absolute zero temperatures of outer space.

Designed by the US-based Clouds Architecture Office, this futuristic project is bound to blow your mind. It proposes a new way of living without ever setting foot on the ground, a concept so revolutionary that even birds might find it unsettling ( Fig. 1 ). The plan involves hanging a skyscraper from an asteroid orbiting Earth. The asteroid will provide a permanent mobile base, supporting the building from space, rather than from the ground, as traditionally done.  The skyscraper will stretch from above the Earth's surface to well beyond the atmosphere ( Fig. 2 ), effectively 'scraping' the air with its base and the vacuum with its top. Fig. 1 . Analemma Tower hangs over New York. Source: Clouds Architecture Office (modified)    Although not feasible with today's technology, the project is grounded in real physics and highlights an important subject of future space constructions. Whether we will ever want to live above the Earth, we are already working there, keeping the International Space Station (ISS) continuously occupied. The design relies on the principles of orbital mechanics, a highly counterintuitive branch of physics. Our life experiences shape our intuition, and for now, only a small number of people have had the opportunity to experience orbital motion. The Analemma Tower design presents an excellent opportunity to explore the workings of orbital mechanics while providing a realistic depiction of what the future may hold. Fig. 2 . Analemma Tower projects well above the Earth's atmosphere. Source: Clouds Architecture Office What's in the name? The name Analemma Tower was selected to reflect the asteroid's orbit, which is the key factor influencing the skyscraper's design. To keep the skyscraper permanently above a localised area on the ground, the asteroid must be in a geosynchronous orbit, which has an orbital period of 24 hours  matching Earth's rotation . If the orbit were equatorial, the skyscraper would stay above a fixed point on the equator. However, the project designers wanted it to hover over New York, where they reside. Consequently, the asteroid's orbit had to be tilted to align with New York's latitude, causing the tower to trace the Analemma pattern ( Fig. 3 ), which inspired its name. Fig. 3 . An Analemma pattern, drawn by the Sun, occurs due to the 23.5-degree tilt of Earth's axis as it rotates around the Sun . By Jfishburn at English Wikipedia, CC BY-SA 3.0. Significance of the geosynchronous orbit.  The geosynchronous orbit is located at an altitude of about 22,236 miles (35,786 km) . Although this altitude poses extraordinary design challenges, requiring the length of cable and tower to match this height ( Fig. 4 ), the choice of a geosynchronous orbit is a necessity. In this orbit, the entire structure, including the asteroid, cable, and tower, would move in sync with Earth's rotation. This alignment ensures that the lower part of the tower moves in tandem with the Earth's atmosphere, significantly reducing atmospheric drag and structural oscillations. A non-synchronous orbit would result in the tower being continuously dragged through the atmosphere at high speed, eventually causing it to be torn apart. Fig. 4 . Analemma Tower is an enormous, self-sustaining complex designed to accommodate all aspects of daily life, including living spaces, offices, farming areas, leisure activities, shops, religious services, and funeral arrangements. Source: Clouds Architecture Office (modified) Why was the asteroid chosen as an anchor? The altitude of a geosynchronous orbit must align with the center of mass of the entire orbiting structure, which includes the anchor, the cable, and the tower. For the orbit to remain stable, the center of mass must be within the anchor's body, which requires the anchor's mass to be significantly greater than the combined mass of the cable and tower. Given the tower's size, a massive rocky asteroid seems to be the most suitable choice. A sufficiently large asteroid will absorb the cable stresses caused by activities within the tower and winds affecting its lower part. The orbital mechanics underlying the project. In orbital motion, there is an ongoing battle between orbital velocity and gravity. When velocity wins, an object moves higher. When gravity takes over, it pulls an object down. Gravity varies with altitude, being strongest at the Earth's surface and diminishing with distance from Earth. In a stable orbit, velocity perfectly counterbalances Earth's gravitational pull at that specific altitude, allowing a satellite to maintain a permanent distance from Earth. This explains why all satellites, including our natural satellite, the Moon, stay in their orbits without falling to Earth. Fig. 5 . The International Space Station (ISS) maintains a stable altitude of 250 miles (400 km) because its orbital velocity of 7.67 km/s (4.75 miles per second) precisely offsets the gravitational field strength of 0.9 g. As altitude increases, gravity decreases, necessitating a slower orbital speed. On the other hand, a lower orbit demands a faster speed to counterbalance stronger gravity. Therefore, whether an object rises, falls, or remains at the same distance from Earth depends on how its orbital velocity compares to the velocity required at that altitude. For example, the International Space Station  (ISS) orbits Earth at an altitude of about 250 miles (400 km),  where gravity is 90% of its strength at Earth's surface ( Fig.5 ). The station maintains a stable orbit because its speed of 17,100 mph (27,600 km/h) precisely balances the gravitational force at that altitude. How does an asteroid support the Tower? An asteroid acts as a counterweight to support the tower, pulling the tower outwards while gravity pulls it downwards. This outward pull is due to centrifugal force, created by the asteroid's circular motion, much like how clothes in a centrifuge are drawn away from its center. But why isn't the entire structure, including the asteroid, cable, and tower, drawn away from Earth, given that all its components rotate together, completing one revolution every 24 hours? This is because, despite having the same orbital period, the structure's components experience different gravitational forces and move at varying orbital velocities. Consequently, they balance speed and gravity differently. The structure will extend vertically for thousands of miles, encountering significant variations in the gravitational field ( Fig. 6 ). At the top, the structure will experience only around 2% of the gravity felt at the bottom. As gravity decreases with altitude, the orbital speed increases. The structure orbits Earth while staying aligned with Earth's radial gravity vector. Consequently, the top of the asteroid must move significantly faster than the base of the tower to traverse a longer distance (tinted red) within the same time frame. Fig. 6 . The various elements of the structure, comprising an asteroid, a cable, and the Analemma Tower, orbit Earth at varying speeds (red-tint). The geosynchronous altitude aligns with the structure's center of mass. The system's component responsible for maintaining geosynchronous altitude and speed is the system's center of mass (COM), situated within the asteroid's body. The COM must travel at a speed of approximately 7,000 mph (11,300 km/h) to accurately counteract the gravity at that altitude, which is only about 2.3% of Earth's surface gravity. Above the COM, gravity weakens, requiring a slower orbital speed, whereas below the COM, gravity intensifies, demanding a faster speed. However, due to the circular motion, the speed changes in the opposite direction, increasing towards the top and decreasing towards the bottom. Consequently, centrifugal force will dominate above the COM, pulling the asteroid upwards. The upward pull will offset the downward pull on the cable and tower, where gravity exceeds centrifugal force below the COM. This tag of war between the centrifugal and gravitational forces will generate significant tension in the cable, maintaining the entire structure taut and balanced. How would your weight change on different floors? If the Analemma Tower's base were close to the ground, its orbital speed would align with Earth's rotational speed, while experiencing the same gravitational strength of g = 9.8 m/s 2 . Therefore, your weight would be the same as on the ground, making you feel as if you are in a conventional, ground-based skyscraper . Even on the 50th floor, your weight would barely change because this altitude has a negligible effect on gravitational strength and tangential speed. This is because the radial lines diverge from the center of the Earth, necessitating an altitude to be compared to the Earth's radius of about 6,400 km (4,000 mi). If the tower's top were near the asteroid, its orbital speed would nearly match that of the asteroid, while experiencing almost the same gravitational strength of 0.023 g. The asteroid's orbital speed is designed to negate the gravitational pull at its altitude, creating a microgravity environment where you feel weightless, much like astronauts on the ISS . The project doesn't specify the tower's length for precise calculations. After all, the Analemma Tower is more of a conceptual idea than a practical proposition. The considerations provided offer a basic understanding of how your weight would change throughout the tower, making you feel grounded on the lower floors and practically levitating on the upper floors. Advantages of space support over ground foundations.  The project is currently unfeasible due to the lack of superstrong materials capable of withstanding the enormous tensile and shear forces affecting a cable and a skyscraper as tall as the Earth's diameter. Still, the idea of supporting such a structure from space is theoretically more achievable than constructing it from the ground up. Structures built on Earth experience compression at their base, which materials handle less effectively than the tension from hanging. A skyscraper as tall as the Analemma Tower would be crushed by its weight because the stress at its base would exceed the strength of even hypothetical materials. So, as surprising as it may be, the concept of suspending a super-tall skyscraper from space is technically more realistic than supporting it from the ground. As we progress in space exploration, this seemingly science fiction concept could offer an alternative for establishing settlements above the inhospitable planets.  Depending on a planet's characteristics, the settlements could be positioned at a lower altitude, reducing the length of the suspended structure and easing the material and engineering requirements. For example, if Earth rotated 16 times faster, the geosynchronous orbit would be around 190 miles (300 km) high, which is just below the ISS orbit. This would dramatically reduce the length of the Analemma Tower, cutting construction challenges by 90%. While this faster rotation would largely resolve the Analemma project's problems, it would also cause chaos on our planet! However, we'll reserve this topic for another discussion.

It's undeniable that black holes have a reputation for being voracious mass eaters. R umors have it that they are destroying our Universe with their appetite, leaving voids in places where galaxies once thrived. With gravity "so strong that nothing can escape", they suck everything in like giant hoovers and won't stop until the last star has gone. Some even claim the dirty deed has been completed, and we already live inside a black hole, unable to escape. But are they truly that malicious or just the victims of malicious gossip? Fig. 1 . A black hole with an accretion disk. Artistic impression. Source: wallpaperflare As the famous saying goes, the rumors about black holes' long-reaching hands are greatly exaggerated. When it comes to gravity, the difference between a black hole and a star of similar mass is significant only within a certain distance from its center. The gravitational field generated by a body is determined by its mass, so celestial bodies with similar mass create similar fields. However, the gravitational field behaves differently above and beneath the body's surface. Above the surface, gravity follows the inverse square law, while below, the distribution is linear. This is where the distinction between a black hole and a star becomes evident, as the black hole's surface contracts to nearly zero. Consider two spherically symmetric celestial bodies, one Yellow and one Grey, as shown in Fig. 2 . The bodies have the same mass: M (yellow) = M (grey). However, the mass of the Grey Body is compressed into a much smaller sphere, with a radius of r that is 1/6 of the Yellow Body's radius R , where R  = 6 r . This size disparity results in the Grey Body's surface being 6 times nearer to its center, leading to a 36-fold increase in surface gravity due to the inverse square law, g(grey) = 36 x g(yellow). Therefore, if you weighed 200 lb (91 kg) on the Yellow Body, your weight would rise to 7200 lb (3276 kg) on the Grey Body. This is the difference you would certainly notice. Fig. 2 . The Yellow and Grey Bodies exert equal force on the Blue Body, because they both have the same mass (M) and their centers are at equal distance (d) from the Blue Body's center . However, if we position a Blue Body at the same distance, d , from the centers of the Yellow and Blue Bodies, each will exert the same gravitational force on the Blue Body. According to Newton's shell theorem, the gravitational field created by a spherically symmetric mass acts as if all its mass is concentrated at its center. This concept allows us to consider planets and stars as point masses when analyzing their orbits. The formula for determining the gravitational field strength, g(d) , produced by mass M  at a distance d , does not involve the mass's radius, the only stipulation being that the distance d  must be outside the mass or on its surface, d ≥ R . Therefore, two bodies with equal mass M but different densities (and thus different radii) will exert equal force on a third body, provided that this third body is equidistant from their centers and located outside both. With all the necessary information at hand, we can now explore how a star transforms into a black hole through gravitational collapse. Gravity is constantly present and always attractive. Without any opposing forces, all matter in the universe would ultimately collapse together. A star maintains its stability as long as the outward pressure from nuclear fusion in its core balances the inward pull of gravity. Once a star exhausts its nuclear fuel, gravity takes over, causing the star to contract rapidly. Let the Yellow body become a star that has depleted its fuel and undergone gravitational collapse, hypothetically without losing mass. When it reaches the stage where its size shrinks to that of the grey body ( Fig. 3 ), its surface gravity increases 36 times in accordance with the inverse square law, as previously discussed. This dramatic rise in surface gravity will continue as the star contracts further.  Once the radius of the collapsing star shrinks to the Schwarzschild radius, the star transforms into a black hole. The Schwarzschild radius marks the spatial boundary where the gravitational field becomes so strong that the escape velocity equals the speed of light, c. Although the star continues to contract, its final state remains unknown. Since nothing can surpass the speed of light, no signal can come back to us after crossing the Schwarzschild boundary to tell the story. Fig. 3 . The Star undergoes gravitational collapse, shrinking first to the size of the grey body, and finally contracting to the black hole. While the Schwarzschild boundary signifies the point of no return for all known particles, this region is minuscule on a cosmic scale.  As one moves away from the event horizon, the gravitational pull of the black hole decreases at the same rate it increased before.   At a distance of d=r  from the black hole's center , the gravitational field weakens to match the Grey Body surface gravity. At a distance of d = R , the field further weakens to equal the Yellow Body surface gravity, which was the surface gravity of the star before the hypothetical collapse. Outside that distance, the fields of both the star and the black hole become identical, continuing to decrease at the same rate. If we placed the Blue planet ( Fig. 4 ) at an equidistance d>R  from the centers of the star and the black hole, they would pull the planet with equal force. Fig. 4 . The Star and the Black hole exert equal force on the Blue planet, because they both have the same mass (M) and their centers are at equal distance (d) from the Blue planet. Our Sun is too small to form a black hole. Still, if we were to replace the Sun with a hypothetical black hole of identical mass, the planets would continue their orbits as if nothing had changed, with the same distances and periods. Without the Sun's energy, the solar system would go dark and freeze. In terms of gravity, differences would arise only in the region between the Sun's radius of ~700,000 km ( ~ 400,000 miles) and the proposed black hole's radius of ~3 km (~2 miles). Surprisingly, these differences would not increase the black hole's ability to capture foreign bodies. In fact, the opposite would happen. While asteroids and comets already bound to the Sun would continue their paths, interstellar wanderers would significantly decrease their chances of colliding with the black hole due to its tiny surface area.. Instead of plunging into the Sun's expansive photosphere, they would now target a 3-km bullseye to meet their end. Potential prey would also avoid the super gravity. Travelling at high speed, they would need to pass within a few kilometers of the black hole for the trap to work, providing the predator with a narrow opportunity in the vastness of the solar system. All in all, a poor score for a famed mass eater. The Sun is doing a better job than this. In the event of gravitational collapse, the Sun would lose its magnetic field, which currently protects it from Galactic Cosmic Rays. This field is generated by the movement of hot plasma inside the Sun. As the plasma breaks down, the field disappears, allowing the interstellar nuclei, protons, and electrons to reach the black hole for consumption. Nevertheless, the dramatic decrease in surface area would negate any advantages. In view of all the evidence, we can conclude that accusations brought to black holes for crimes against the masses look more like character assassination than fair justice! Black holes fascinate us with their invisible yet formidable presence. They evoke visions of colossal cosmic phantoms, shrouded in mystery and fraught with danger. We fear most what remains unseen. Their unique ability to capture light, which is essential for our vision and measuring devices to access data, sets them apart from other celestial bodies and endows them with magical properties they do not actually have. We may never discover what exists inside black holes, as any signal that passes through the event horizon boundary gets obliterated from our data. Still, the black holes don't trap gravity, allowing us to study their gravitational fields just as we do with other masses. Therefore, we can confidently assert that their gravitational advantage is confined to regions near the event horizons. Outside of these areas, their gravitational pull is no different than that of other celestial bodies with a similar mass.

Mirrors love messing with our heads, challenging us with deceptive illusions that can make us question our sanity. Being accomplished tricksters , they can leave us at the end of our wits, wondering, "How is this even possible?" One example is when a mirror demonstrates a spooky side by reflecting an object it shouldn't be able to see. In Fig. 1 , a figurine is hidden from the mirror's view by a piece of paper. Yet, the mirror reflects the figurine, as if it had gone around the paper and taken a cheeky peek a t what is on the other side. Fig. 1 .  A mirror reflects the figurine that appears to be hidden from its view by a piece of paper. Source: ILF Science A mirror creates images by reflecting light off its surface. While light doesn't pass through the surface, images appear to extend behind it. This optical deception is so convincing that our brain often struggles to differentiate between reality and illusion. In Fig. 1 , a mirror deceives us into believing that a reflection extends beneath the figurine, when in fact, the reflection is formed on the mirror surface, placed at a 90° angle to the figurine. Let's replace the figurine with a cuboid for simplicity and ignore, for a minute, a virtual image of the cuboid that our brain will create underneath the paper ( Fig. 2 ). Here, we trace a trajectory of the light rays bouncing off the cuboid. Note that a ray reflected off the cuboid's top l ands on the mirror farther away than the one reflected off its middle . These rays will next bounce off the mirror into the eye, contributing to the formation of a virtual image of the top and mid parts of the cuboid. The rays bouncing off the cuboid's base (not shown) will land closer to the cuboid, hitting the paper, thus blocking the mirror from 'seeing' the base. Fig. 2 .  The ray reflected from the cuboid's top lands farther away from the cuboid than the ray reflected from its mid part. Our ability to perceive objects and their mirror images is based on the same principle outlined in the law of reflection. Mirrors merely redirect the view we would see from their position to our eyes. This is why we perceive two objects in different locations: one real, which we see directly, and one illusory, which we would see from the mirror's position. Mirrors are unique in that they are designed to preserve the properties of the light they reflect. They don't alter the light's characteristics, such as wavelengths and angles of incidence. This unique feature allows them to replicate the real world so convincingly that many cultures attribute them supernatural power. Since mirrors don't interfere with the light properties, keeping them intact, they act like phantoms, deceiving the brain into thinking that the reflected light comes directly from the object, with nothing standing between the object and the eye. The brain overlooks the presence of the mirrors ( Fig. 3 ) and projects the light rays backward, much like it does when light comes directly from the object. As a result, a virtual image appears behind the mirror, identical to the real one, as it is constructed using the same information that defines the real image. Fig. 3 .  A virtual image of the cuboid is created due to our brain extending reflected light rays backward to end behind the mirror. Mirrors are exceptional in their capacity to reflect light without altering the fundamental information carried by the rays. This feature is responsible for all mirror illusions. No other surface can equal a mirror in this ability. For example, placing the figurine on a polished steel surface would still produce a reflection, but its quality would be compromised because steel absorbs certain wavelengths and changes the original angles of incidence. Rough surfaces, like paper, perform even worse, destroying the original data by scattering wavelengths in all directions. The only information our eyes obtain from light reflected off the paper pertains to the paper itself. On such a surface, the figurine can only cast a shadow, which will stretch across the surface, clearly indicating where it belongs. The paths of rays, hitting the figurine and reflected off it, are mapped in Fig. 4 . Rays bouncing off the two points on the figurine's lower part land on the paper and are perceived as two points on the paper, because the information of the figurine's points is lost. Rays bouncing off the two points on the figurine's upper part land on the mirror and are perceived as the two points on the figurine, because the mirror doesn't manifest its presence and doesn't feed its personal information into the rays it reflects. If we removed the paper, a complete image would emerge, with the points on the paper matching those on the figurine. The mirror image would join the original at the feet and appear to extend beneath it, despite being stretched over the mirror surface. This optical illusion deceives us into thinking the figurine is hidden from the mirror's view, whereas in reality, the mirror receives complete information about the parts it displays. Fig. 4 .   When the paper is removed, the points underneath it will match those on the figurine.

We expect a table to have legs. After all, how else can it support the top in its position? The idea of replacing legs with chains seems absurd. We use chains to suspend a chandelier from the ceiling. And we can't suspend a table from the floor unless, by some magic, we reverse the direction of gravity. Yet, anti-gravity tables appear to do just that, giving the impression of floating in the air ( Fig. 1 ). The trick is that their clever design makes gravity keep a tabletop elevated by pushing it down. Fig. 1    The anti-gravity table looks like it is floating in the air. Gravity is always strongest at the Earth's surface, pulling all objects toward it. A chandelier may look like it is supported by a chain hanging from the ceiling. But in fact, support comes from the walls resting on the ground. Similarly, all table's components are supported by the lower part touching the floor ( Fig. 2 ), with the top hanging from it like a chandelier from the ceiling, with one distinction. For chandeliers, all their parts are positioned below the ceiling attachment point. In our case, however, the section above the chain makes the equilibrium unstable. Fig. 2 The top section of the anti-gravity table is suspended from the bottom section by a chain. The instability problem is resolved with the help of the remaining four chains. The two chains on the right ( Fig. 3 ) stop the left side of the tabletop from dropping to the floor. Likewise, the two chains on the left will keep the right side elevated. This combination completes the design, enabling the structural stability of the table as a functional piece of furniture. Fig. 3    Two chains attached to the right end secure the elevated position of the left end. The weight of the tabletop is counterbalanced by the tension in the chains, which are carefully adjusted in length to ensure maximum stability. This concept, known as tensional integrity or tensegrity, is used in engineering to create stunning architectural structures that appear to defy gravity. One example is the Kurilpa Bridge in Australia, the world's largest tensegrity bridge with stainless steel tensegrity cables  ( Fig. 4 ) . This jewel of modern engineering also features a sophisticated solar-powered lighting system programmed to illuminate it in various colors at night. Fig. 4   Kurilpa Bridge in Brisbane ( Queensland , Australia) is the world's largest hybrid tensegrity bridge. Source: Wikipedia

Puzzle.   You have a glass of milk and a bucket of water. First, take a spoonful of milk from the glass and add it to the bucket. Then, take a spoonful from the bucket and add it to the glass. Which will be greater: the amount of water in the glass or the amount of milk in the bucket?    Answer.  Equal  Fig. 1 A glass of milk and a bucket of water. Solution.  The result is unexpected because we added a spoonful of milk to the bucket ( Fig. 1 ) and moved less than a spoonful of water to the glass. However, this spoon, which is partly filled with water, must also contain milk from the bucket to complete it. We use the same spoon for both actions. This guarantees that the amount of foreign liquid in the containers (water in the glass and milk in the bucket) will be equal. Solution.  Nevertheless, this spoon, partially filled with water, must also include milk from the bucket to fill it completely. We use the same spoon for both tasks. This ensures that the quantity of foreign liquid in the containers (water in the glass and milk in the bucket) will be the same. I find the proof by contradiction most effective so that we will apply it here. Suppose the milk glass has more water than the water bucket has milk. This implies that the glass gained more liquid than it lost, increasing its volume. Conversely, the bucket received less than it lost, causing a decrease in the liquid volume. However, we used the same spoon in both transfers, so the liquid volume should stay unaffected. We arrived at the contradiction. Therefore, our original assumption is incorrect, and the amount of foreign liquid in the containers must be equal. We can also use numbers to illustrate the answer. Let the spoon from the bucket contain  1/3 water and 2/3 milk  ( Fig. 2 ). Then, we removed 2/3 milk from the bucket, leaving behind 1/3 milk, and added 1/3 water to the glass. As a result, both containers end up with an equal amount of foreign liquid, circled in red: 1/3   water in the glass and 1/3   milk in the bucket. Fig. 2 The milk glass has 1/3 water, and the water bucket has 1/3 milk after the back-and-forth liquid transfer. Let the spoon from the bucket hold 6/7 water  and 1/7 milk  ( Fig. 3 ). Then, we removed 1/7 milk from the bucket, leaving 6/7 milk behind, and added 6/7 water to the glass. Consequently, both containers again end up with an equal amount of foreign liquid: 6/7   water in the glass and 6/7   milk in the bucket. The key is that what is native in one container becomes foreign in the other . So, the amount of local liquid ( water ) removed from the bucket will always match the amount of foreign liquid ( milk ) left behind. Fig. 3 A spoon from the bucket adds 6/7 water to the glass, leaving 6/7 milk in the bucket. It will make no difference whether we stir the bucket or not. If we scoop all the milk from the bucket, we transfer no water to the glass, leaving both containers free of foreign liquid. If we scoop no milk, a spoonful of water will be transferred to the glass, resulting in both containers having a spoonful of foreign liquid. We have already examined the scenario where the spoon is partly filled with milk and water.   The size of the containers is also irrelevant. The milk container can be larger than or the same size as the bucket .   The crucial point is to use the same spoon for both actions , ensuring the amount of liquid transferred back and forth stays the same. Then:   3/3 milk (moved to bucket) = 1 full spoon = 2/3 milk + 1/3 water and 7/7 milk = 1 spoon = 6/7 water   +   1/7 milk

Observing the scenery from the top of a skyscraper can be an uplifting experience. Many skyscrapers offer stunning panoramic views that extend across the city and into the distance. As you ascend higher, the horizon expands, unfolding countryside and distant towns miles away. This breathtaking experience can make you feel like a bird gliding in the air, with gravity easing its grip on your body. But is this sensation deceptive, or do we indeed weigh less at the top of a skyscraper? Fig. 1 . Viewing scenery from the skyscraper can make you feel like a bird gliding in the air. What is weight? To answer this question, let's delve into the physics of gravity and understand how it influences our weight. Weight, W, is determined by the mass of our body, m, and the strength of gravity at our altitude, g, W=mg. Mass remains constant regardless of where we are: on the ground, on the top of the mountain, or even on another planet. However, the gravitational field varies with altitude, strongest at the Earth's surface and weakening as we move away from it. Therefore, it is true that we weigh less at the top of the tall building. The question is whether this difference is significant enough to be detected by a regular scale. Why does weight decrease with altitude? Gravity weakens as altitude rises due to the geometric properties of our three-dimensional space. The gravitational field is radial ( Fig. 2 , left), meaning it gets diluted as the distance from the source of gravity increases because the radial lines diverge ( Fig. 2 , right). The rate of this divergence is governed by the inverse square law, which states that the strength of the gravitational field is inversely proportional to the square of the distance from the source of gravity. The radial lines converge at the Earth's center. Consequently, the distance is measured from that point. Fig. 2. The Earth's gravitational field weakens with distance from Earth, according to the inverse square law. The gravitational field strength at a specific altitude is measured  by g, which determines how fast an object will accelerate if dropped at that level. If we drop two identical stones, one near the ground and the other from the peak of Mount Everest, the stone dropped near the ground will accelerate faster because it's closer to Earth's center. Nevertheless, the difference is minimal, only 0.4%, as Everest's height is minor compared to Earth's radius. The reduction in gravity, and consequently in weight, becomes significant when the altitude is comparable to Earth's radius. At ground level, we are one radius R , or 4,000 miles (6,400 km) from the Earth's center ( Fig. 3 ), experiencing a gravitational acceleration of g = 9.8 m/s 2 . If we double this distance, placing ourselves one radius R  above the Earth's surface, we will  experience an acceleration that is a quarter of the ground value . That, if you weighed 200 pounds (91 kg) on the ground floor of a hypothetical skyscraper, reaching 4,000 miles (6,400 km) above Earth, you would weigh only 50 pounds (23 kg) on its top. This is a dramatic weight loss, but so is the building's height, unachievable with modern technologies. Fig. 3 . At a distance of 2 radii (2R) from Earth's center, our weight is a quarter of that at the Earth's surface. Earth's size makes skyscrapers appear small. So, just how tall are these skyscrapers? Currently, the Burj Khalifa in Dubai holds the record, reaching 2,717 feet (828 meters). When you compare this height to the Earth's radius,  the difference in weight becomes insignificant and imperceptible on a normal scale. The size of Earth dwarfs everything in its vicinity; from an airplane, even mountains appear flat. The skyscraper would have to reach above the Earth's atmosphere to exhibit a noticeable difference in weight. At the altitude of the International Space Station (ISS), which is 250 miles (400 km), the gravity is still 90% as strong as it is on the Earth's surface. The microgravity experienced onboard is due to the balance between gravity and orbital speed. If a tower could be constructed to reach this astonishing height, our weight would be reduced by 10%. To put it in perspective, if you weigh 200 pounds (91 kg) on the Earth, you would weigh 180 pounds (82 kg) at the top of such a tower. This is a significant change, but who would want to live above a breathable atmosphere? The microgravity experienced onboard is due to the balance between gravity and orbital speed. If a tower could be built to reach this remarkable height, our weight would decrease by 10%. For example, if you weigh 200 pounds (91 kg) on Earth, you would weigh 180 pounds (82 kg) at the top of such a tower. This is a notable difference, but who would want to live above a breathable atmosphere?

We find ourselves in ancient Greece, around 250  BC. King Hiero   of Syracuse commissioned  local goldsmiths to make a crown and got suspicious that the gold was mixed with silver. The formidable ruler vowed to catch the culprits and asked Archimedes, the famous mathematician and physicist, to help. Gold is nearly twice as dense as silver. Therefore, a gold crown would have a smaller volume than one mixed with silver, assuming they both had the same weight. Estimating the volume of a regular object posed no issue. B ut how could one determine the volume of the intricate crown without melting it down? That was the challenge Archimedes faced. Fig. 1 . Finding the volume of an irregular crown was the challenge Archimedes faced.   Legend has it that  Archimedes  discovered a solution while immersing himself in the bath ( Fig. 2 ). Observing the water level rise, he realized that   the displaced water must match the volume of his submerged body. Thus, by submerging the crown in water, he could calculate its volume by measuring the change in water level. The elegant simplicity of this solution made him forget his nudity and run to Hiero's palace, exclaiming "Eureka!" or "I've found it!" This naked dash became a historical testament to the immense joy a discovery can bring to a scientist. Fig. 2   Archimedes in a public bathhouse.   Woodcut, 16th century. Source: alamy.com Equipped with this innovative method for determining the volume of irregular objects, Archimedes carried out the following experiment. He balanced the crown with pure gold pieces to ensure equal weight. If the crown contained silver, it would be larger and displace more water. He then placed the gold pieces in the water container and marked the water level. Afterwards, he put the crown in the container and observed that the water rose above the mark. The goldsmiths were caught cheating; science triumphed over greed. Tragically, Archimedes was killed during the Roman invasion. Syracuse is now part of the island of Sicily in Italy.   His final words to his  assailant were, "Please, don't disturb my papers!"   The brightest mind of his time, he lived and died like a legend, devoted to science  until his last breath. Some of his legacy was lost during the war. His surviving work, the treatise 'On Floating Bodies', led to further developments in fluid dynamics, culminating in the formulation of the Archimedes principle as we know it today: a body immersed in a fluid experiences a buoyant force equal to the weight of the fluid it displaces . Today, by applying the buoyancy principle, we can significantly simplify the test, letting nature do the math. After balancing the crown with pure gold  in the empty container ( Fig. 3 , left), we can fill it with water  (Fig. 3 , right). If the crown is pure, the scale will remain in balance. However, if the balance tips towards the gold, the crown has a larger volume, indicating the presence of silver. Fig. 3. The advanced method of crown testing uses the buoyancy principle. Assume the crown is pure and has a volume of 4 cm³, implying the gold must also be 4 cm³.  In the empty container, t he buoyant force will be the weight of 4 cm³ of air, identical for both. In the filled container, t he buoyant force will be the weight of 4 cm³ of water, again the same for both. The scale will stay balanced in any medium because the force pushing them up will remain equal due to the equality of their volume. Now, let the crown be impure, with an additional  volume of 1   cm ³, highlighted in red in Fig. 3 . When the scale is balanced in the empty container, it accounts for the additional upward force, equal to the weight of 1 cm³ of air. In the filled container, this extra force increases because 1 cm³ of water weighs more than 1 cm³ of air. The resulting increase in buoyancy will lift the crown, causing the scale to tip towards the gold. We can reverse the test and balance the crown in water first, which will necessitate the removal of some gold from the right pan. Once the water is drained, the crown will outweigh the gold because the buoyant force that supported it in water decreases in the air. This test can be repeated in any medium and any sequence. The balance will be disrupted when transferring the scale from one medium to another, as different media exert varying buoyant forces.

Problem: Two cyclists are approaching each other with speeds of 2 mph and 3 mph. When they are 10 miles apart, a bird starts flying back and forth between them at a speed of 12 mph, maintaining this pattern until the cyclists meet. What distance will the bird travel? Answer: 24 miles Fig. 1 . Two cyclists approaching each other with a bird flying between them.. Solution. If you think this problem requires complex math , you're in for a pleasant surprise. This is more of a logic problem than a mathematical one, meaning it can be solved with basic arithmetic in two simple steps. Its elegant solution highlights a relationship between speed, distance, and time, three fundamental concepts used in physics. If you understand how this relationship works , you will find this problem a doddle. A bird joins the cyclists on their journey when it aligns with the first one, as shown in Fig. 1 . This moment marks the distance between the cyclists becoming 10 miles and initiates the countdown to the end of their shared journey. Since all three start and finish at the same time, the time they spend on the trip is the same. This is the crucial factor that will lead to a quick solution. We can reverse the bird's direction and align it first with cyclist B. This won't alter the outcome because it will change neither the cyclists' speed nor the distance between them, so that the travel time will remain the same. Now that we have established that the travel time is the same for everyone, we can calculate it using the closing speed formula. The closing speed is the rate at which two cyclists close the gap between them. Assuming D   is the distance between cyclists, V1 and V2 are their speeds, we estimate the travel time as: T = D ÷ (V 1 + V 2 ) = 10 miles ÷ 5 mph = 2 hours Knowing the bird's speed, V3 , and the travel time, T , we can find the distance, d , it travels: d = V3 x T = 12 mph x 2 hours = 24 miles The bird will cover a distance of 24 miles.   The puzzle's solution, which otherwise would require calculating the infinite series of back-and-forth trips, is commonly credited to Hungarian mathematician Paul Erdos. A condition is applied to the puzzle: the bird's speed can't be less than the cyclists'. If it were slower, the faster cyclist would disrupt the bird's free flight and drag it along until reaching the other cyclist. This would prevent the bird from moving back and forth at its speed, compromising the puzzle's integrity and reducing it to a regular problem with a predictable solution.

Problem. A bat and a ball cost $1.10 together. The bat costs $1 more than the ball. How much is the ball? Answer. The ball costs $0.05. Fig. 1. A bat and a ball problem Solution. While not difficult, this puzzle often leads to incorrect answers due to psychological factors. It gained fame through Daniel Kahneman, an expert in the psychology of judgment and decision-making, who featured it in his best-selling book "Thinking, Fast and Slow". The book examines human behavior by analyzing how our brain processes information. Kahneman explains that our brain operates in two modes: fast and slow. The fast mode is always active, permanently running in the background, to help us deal with routine tasks and respond swiftly in case of an emergency. The slow mode engages when we pause to evaluate the situation and find the best solution. Put it this way, the first mode can be described as intuitive, while the second is analytical. The apparent simplicity of the puzzle triggers a fast response, resulting in the wrong conclusion. We see two numbers, $1.10 and $1, which can be easily subtracted, and we know we need to determine the difference between specific values. The brain picks up the easy clues and deduces that the ball must cost 10 cents. However, the ball is $1 cheaper than the bat, not their combined cost. This key detail is hidden from view by the clever choice of numbers and the puzzle's deceptive setup. If you also fell into this trap, don't worry. You are in good company. A group of Harvard students made the same mistake when pressed for a prompt response, even though they could handle far superior conundrums. Our fast mode picks up superficial clues, which can mislead us into making wrong decisions.   The puzzle was created by Yale professor Shane Frederick, an expert in consumer behavior, to illustrate how skillful wording can shape our perception of a product and affect our spending habits. We are all guilty of buying things we don't need, swayed by tempting offers like 'Buy One Get One Free'. Another example of consumer manipulation involves pricing tags such as $59.99 instead of $60. Saving a single cent while spending tens of dollars can't improve our finances. Still, these strategies work, and we can't deny it. There is a science behind our psychological traits, which marketing professionals use to encourage impulsive buying and assure us we've made the right choice.   I will rephrase the puzzle to correct the misleading wording, and the solution will become clear: A bat and a ball cost $1.10 together. The difference between their prices is $1.00. How much is the ball? We now see we have two unknowns and two equations to solve. I will employ visual object recognition to streamline the solution. This tactic is commonly used in consumer marketing and educational strategies, albeit for different purposes. Your education helps you withstand manipulations and form independent opinions that benefit you and not others. The two equations are as follows: The correct answer is that the ball costs $0.05 , and the bat costs $1.05 .

Puzzle: Three students, Emma, Peter, and James, shared a pizza. Emma ate a quarter, Peter ate a third , which resulted in Peter having 2 slices more than Emma. How many slices did James have? Answer:  James had 10 slices   Fif. 1 Emma, Peter, and James shared a pizza. Solution:  Emma and Peter's shares are given as fractions (1/4 and 1/3). This allows us to calculate James's share because the sum of all shares must amount to one pizza, Fig. 2 , left.   1/4 + 1/3 = 3/12 + 4/12 = 7/12 (Peter and Emma's shares combined) 12/12 - 7/12 = 5/12 (James's share) Fig. 2 Peter's share is bigger than Emma's by 1/12 (left) or by 2 slices (right) The difference between Emma's and Peter's shares is given as 2 slices. This information helps us determine the number of slices in a single fraction ( Fig. 2 , right). The difference in shares between Peter and Emma is:   4/12 - 3/12 = 1/12   By converting James's share (5/12) into slices at a rate of 2 slices for each 1/12 of the pizza, we get: Fig. 3 Converting 5/12 into slices at a rate of 2 per 1/12 gives 10 slices The pizza was cut into 24 slices. To determine each student's share, we split it into 12 parts, each containing 2 slices. Emma consumed 3 parts, 3 parts x 2 slices   =  6 slices. Peter consumed 4 parts, 4 parts x 2 slices   =  8  slices, having 2  slices more, 8 - 6 = 2 . James ate 5 parts, 5 parts x 2 slices   =  10  slices.