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Cliff Pickover

@pickover • 232,112 subscribers

Increase your sense of wonder. (Author of 50 books & 800 patents. Yale Ph.D.) "Pickover contemplates realms beyond our known reality." ~NY Times

Shorts

Mathematics and Mystery. Fourier Decomposition of a Pentagram. By George Savva, Used with permission.

Mathematics and Mystery. Fourier Decomposition of a Pentagram. By George Savva, Used with permission.

53,518 просмотров

Engineering, mathematics. Unconventional gears. The center of a logarithmic spiral traces a straight line as it is rolled. By Matt Henderson, Matt Henderson, used with permission.

Engineering, mathematics. Unconventional gears. The center of a logarithmic spiral traces a straight line as it is rolled. By Matt Henderson, Matt Henderson, used with permission.

1,467,453 просмотров

Math. A rare peek from within a "Gyroid," a fascinating minimal surface in geometry that separates space into two oppositely congruent labyrinths of passages. By Andrea Belloni, Water flowing | anbello.eth | anbello.tez, Used with permission. This infinitely extending "labyrinth" surface divides all of space into two identical, intertwined, mirror-image passages: without ever intersecting itself. Discovered by mathematician Alan Schoen in 1970, it's a triply periodic minimal surface (zero mean curvature everywhere, like a soap film), with no straight lines or mirror symmetry. It can be approximated by the surprisingly simple equation: sin(x)cos(y) + sin(y)cos(z) + sin(z)cos(x) = 0. Nature loves it too: gyroid nanostructures in butterfly wing scales create iridescent colors through light interference, and similar forms appear in cell membranes, polymers, and even ketchup-like fluids. It's a superstar in materials science for lightweight, strong 3D-printed structures and photonic crystals.

Math. A rare peek from within a "Gyroid," a fascinating minimal surface in geometry that separates space into two oppositely congruent labyrinths of passages. By Andrea Belloni, Water flowing | anbello.eth | anbello.tez, Used with permission. This infinitely extending "labyrinth" surface divides all of space into two identical, intertwined, mirror-image passages: without ever intersecting itself. Discovered by mathematician Alan Schoen in 1970, it's a triply periodic minimal surface (zero mean curvature everywhere, like a soap film), with no straight lines or mirror symmetry. It can be approximated by the surprisingly simple equation: sin(x)cos(y) + sin(y)cos(z) + sin(z)cos(x) = 0. Nature loves it too: gyroid nanostructures in butterfly wing scales create iridescent colors through light interference, and similar forms appear in cell membranes, polymers, and even ketchup-like fluids. It's a superstar in materials science for lightweight, strong 3D-printed structures and photonic crystals.

41,176 просмотров

Geometry from another world. "The universe buries strange jewels deep within us all, and then stands back to see if we can find them." - Elizabeth Gilbert, Big Magic Art by Joseff, Used with permission.

Geometry from another world. "The universe buries strange jewels deep within us all, and then stands back to see if we can find them." - Elizabeth Gilbert, Big Magic Art by Joseff, Used with permission.

15,689 просмотров

Mathematics. Two spheres, rolling on a strange track, for an eternity. By Matthew Hughes, Matthew Hughes, Used with permission.

Mathematics. Two spheres, rolling on a strange track, for an eternity. By Matthew Hughes, Matthew Hughes, Used with permission.

19,446 просмотров

Mathematics and mystery. "Two Ellipses." By Idan Tal, Idan Tal, Used with permission

Mathematics and mystery. "Two Ellipses." By Idan Tal, Idan Tal, Used with permission

53,426 просмотров

World map shows global population distribution by latitude and longitude. (Will this be very different in 50 years?) By Engaging Data, Used with permission.

World map shows global population distribution by latitude and longitude. (Will this be very different in 50 years?) By Engaging Data, Used with permission.

85,155 просмотров

Visualization by Andrea Belloni, Water flowing | anbello.eth | anbello.tez, Used with permission. Physicist Roger Penrose wrote: "Everything in the physical Universe is indeed governed in completely precise detail by mathematical principles — perhaps by equations... or perhaps by some future mathematical notions fundamentally different from those which we would today label by the term ‘equations’.... Even our own physical actions would be entirely subject to such ultimate mathematical control, where ‘control’ might still allow for some random behaviour governed by strict probabilistic principles." ~ The Road to Reality Belloni visualization employs 256K particles.

Visualization by Andrea Belloni, Water flowing | anbello.eth | anbello.tez, Used with permission. Physicist Roger Penrose wrote: "Everything in the physical Universe is indeed governed in completely precise detail by mathematical principles — perhaps by equations... or perhaps by some future mathematical notions fundamentally different from those which we would today label by the term ‘equations’.... Even our own physical actions would be entirely subject to such ultimate mathematical control, where ‘control’ might still allow for some random behaviour governed by strict probabilistic principles." ~ The Road to Reality Belloni visualization employs 256K particles.

28,834 просмотров

Mathematics, algorithms, spirals, geometry, programming. "The Hidden Music of Concentric Souls." Rotational Moiré Interference from Phase-Shifted Polar Sources. Animation by A. L. Crego, Used with permission

Mathematics, algorithms, spirals, geometry, programming. "The Hidden Music of Concentric Souls." Rotational Moiré Interference from Phase-Shifted Polar Sources. Animation by A. L. Crego, Used with permission

72,517 просмотров

Mathematics. This is a walkable trefoil knot. © Dr. Robert Fathauer, Robert Fathauer, Used with permission,

Mathematics. This is a walkable trefoil knot. © Dr. Robert Fathauer, Robert Fathauer, Used with permission,

14,720 просмотров

Life on the Edge. Creatures from mathematics. We are Not Alone. Experimenting with simulated ecosystems. By SPACEFILLER, Used with permission.

Life on the Edge. Creatures from mathematics. We are Not Alone. Experimenting with simulated ecosystems. By SPACEFILLER, Used with permission.

17,859 просмотров

Mathematics. By Daniel Mentrard, Daniel Mentrard, Source: Used with permission.

Mathematics. By Daniel Mentrard, Daniel Mentrard, Source: Used with permission.

52,351 просмотров

Math and the Mind. Dodecahedron. By Andrea Belloni, Water flowing | anbello.eth | anbello.tez, Used with permission. In your personal opinion, do dodecahedrons exist, or are they abstract ideal geometries, Platonic solids that exist in some non-human realm? Perhaps they exist in both ways: as precise, abstract mathematical ideals and as real physical objects you can hold, study, or even find in ancient ruins and nature. The abstract, ideal version a convex polyhedron with 12 identical regular pentagonal faces, 20 vertices, and 30 edges, where the same number of faces meet at each vertex (all faces congruent, all angles equal, etc.). Platonists argue these forms inhabit an abstract realm independent of human minds or the physical universe. Others see them as human inventions or patterns inherent to logic and reality itself. [Humans have been building dodecahedrons for millennia, and close approximations appear, such as pyritohedrons in pyrite (fool's gold) crystals.]

Math and the Mind. Dodecahedron. By Andrea Belloni, Water flowing | anbello.eth | anbello.tez, Used with permission. In your personal opinion, do dodecahedrons exist, or are they abstract ideal geometries, Platonic solids that exist in some non-human realm? Perhaps they exist in both ways: as precise, abstract mathematical ideals and as real physical objects you can hold, study, or even find in ancient ruins and nature. The abstract, ideal version a convex polyhedron with 12 identical regular pentagonal faces, 20 vertices, and 30 edges, where the same number of faces meet at each vertex (all faces congruent, all angles equal, etc.). Platonists argue these forms inhabit an abstract realm independent of human minds or the physical universe. Others see them as human inventions or patterns inherent to logic and reality itself. [Humans have been building dodecahedrons for millennia, and close approximations appear, such as pyritohedrons in pyrite (fool's gold) crystals.]

17,425 просмотров

Mathematics, motion, polygons, alignment, harmony. The journey begins. By dave, Used with permission.

Mathematics, motion, polygons, alignment, harmony. The journey begins. By dave, Used with permission.

136,475 просмотров

Mathematics and beauty. "It's called the Lorenz 𝘢𝘵𝘵𝘳𝘢𝘤𝘵𝘰𝘳 because all nearby points are 𝘢𝘵𝘵𝘳𝘢𝘤𝘵𝘦𝘥 into the set of chaotic orbits, regardless of initial conditions." By Ben Bartlett, Ben Bartlett, Source: Used with permission.

Mathematics and beauty. "It's called the Lorenz 𝘢𝘵𝘵𝘳𝘢𝘤𝘵𝘰𝘳 because all nearby points are 𝘢𝘵𝘵𝘳𝘢𝘤𝘵𝘦𝘥 into the set of chaotic orbits, regardless of initial conditions." By Ben Bartlett, Ben Bartlett, Source: Used with permission.

119,183 просмотров

Mathematics. Procedural animation. Harmonic tango. By Etienne Jacob, Etienne Jacob, Source: Used with permission.

Mathematics. Procedural animation. Harmonic tango. By Etienne Jacob, Etienne Jacob, Source: Used with permission.

35,796 просмотров

Mathematics. Fractals. → Sierpiński pyramid. Infinity in your hand. Credit: I didn’t upload this video to 𝕏. I’m pointing to a video residing in the 𝕏 stream of “creapills” (Jul 18, 2020). [math, maths]

Mathematics. Fractals. → Sierpiński pyramid. Infinity in your hand. Credit: I didn’t upload this video to 𝕏. I’m pointing to a video residing in the 𝕏 stream of “creapills” (Jul 18, 2020). [math, maths]

31,564 просмотров

This is a mathematics joke. Source:

This is a mathematics joke. Source:

75,496 просмотров

Mathematics. Fractals. Mandelbrot. Infinity. Created by iterating simple formulas, revealing the mystery of math. How chaos theory and iteration can birth entire universes that look alive. By Trips, Used with permission.

Mathematics. Fractals. Mandelbrot. Infinity. Created by iterating simple formulas, revealing the mystery of math. How chaos theory and iteration can birth entire universes that look alive. By Trips, Used with permission.

14,271 просмотров

$Mathematics. A close look at the infinite swirling of the Rössler Attractor, which exhibits chaotic dynamics and fractal properties. Like a cosmic taffy-pulling machine. By Marcus Volz, Used with permission.$

Mathematics. A close look at the infinite swirling of the Rössler Attractor, which exhibits chaotic dynamics and fractal properties. Like a cosmic taffy-pulling machine. By Marcus Volz, Used with permission.

17,598 просмотров

Videos

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Strange Universe. Math, Physics, Reality. "6 belts are anchored on the outside to a fixed frame, and on the inside to a spinning cube. Although the cube spins continuously in one direction the belts never become tangled. The belts return to their original configuration when the cube has rotated 720 degrees." - Jason Hise,

Strange Universe. Math, Physics, Reality. "6 belts are anchored on the outside to a fixed frame, and on the inside to a spinning cube. Although the cube spins continuously in one direction the belts never become tangled. The belts return to their original configuration when the cube has rotated 720 degrees." - Jason Hise,

619,755 просмотров • 3 дней назад

Mathematical Coalescence. Math machine. A strange pattern emerges in these Archimedean Spirals with interconnecting tubes. "The pattern 東南西北 can be displaced, but remains conserved." By Raven Kwok, Raven Kwok, Used with permission

Mathematical Coalescence. Math machine. A strange pattern emerges in these Archimedean Spirals with interconnecting tubes. "The pattern 東南西北 can be displaced, but remains conserved." By Raven Kwok, Raven Kwok, Used with permission

165,001 просмотров • 14 дней назад

Mathematics, physics. We live in a chaotic, unpredictable universe. This is a grid of 812 double pendulums, each with slightly different initial conditions for the two angles of the pendulum. By Jonathan Nafziger, Used with permission

Mathematics, physics. We live in a chaotic, unpredictable universe. This is a grid of 812 double pendulums, each with slightly different initial conditions for the two angles of the pendulum. By Jonathan Nafziger, Used with permission

1,842,256 просмотров • 3 лет назад

Mathematics. Epicycles. Fourier series. Polar bear synthesis. By Daniel Mentrard, Daniel Mentrard, Used with permission.

Mathematics. Epicycles. Fourier series. Polar bear synthesis. By Daniel Mentrard, Daniel Mentrard, Used with permission.

67,558 просмотров • 1 месяц назад

Math, physics, ultimate reality. Don't you wish you could snap your fingers and magically gain a complete and profound understanding of these Heat, Wave and Schrödinger equations? By Daniel Mentrard, Daniel Mentrard, Used with permission.

Math, physics, ultimate reality. Don't you wish you could snap your fingers and magically gain a complete and profound understanding of these Heat, Wave and Schrödinger equations? By Daniel Mentrard, Daniel Mentrard, Used with permission.

21,878 просмотров • 21 дней назад

Mathematics clock. By Daniel Mentrard, Daniel Mentrard, Used with permission. Source:

Mathematics clock. By Daniel Mentrard, Daniel Mentrard, Used with permission. Source:

107,603 просмотров • 5 месяцев назад

$Beauty of mathematical chaos and infinite complexity. Mandelbulb: higher-dimensional extension of the Mandelbrot Set. Central floral-like formation with radiating, recursive details are characteristic of fractals. By Trips, Used by permission.$

Beauty of mathematical chaos and infinite complexity. Mandelbulb: higher-dimensional extension of the Mandelbrot Set. Central floral-like formation with radiating, recursive details are characteristic of fractals. By Trips, Used by permission.

21,100 просмотров • 24 дней назад

Math: Strange Infinite Universe. Using mathematics to create perfect circles from whip-like tendrilous spirals. Infinite sums. By Tungsteno, Tungsteno, Used with permission. Info:

Math: Strange Infinite Universe. Using mathematics to create perfect circles from whip-like tendrilous spirals. Infinite sums. By Tungsteno, Tungsteno, Used with permission. Info:

16,178 просмотров • 23 дней назад

Astronomy, mathematics, physics. The evolution in our thoughts and models: Heliocentrism and Geocentrism. By Malin Christersson, source: (CC BY-NC-SA 4.0)

Astronomy, mathematics, physics. The evolution in our thoughts and models: Heliocentrism and Geocentrism. By Malin Christersson, source: (CC BY-NC-SA 4.0)

470,087 просмотров • 3 лет назад

Roger Penrose's cyclic universe theory, called Conformal Cyclic Cosmology (CCC), proposes the universe goes through infinite cycles, where the end of one "aeon" (a universe's lifespan) smoothly transitions into the Big Bang of the next. Art: Andrea Belloni, Water flowing | anbello.eth | anbello.tez, Used with permission

Roger Penrose's cyclic universe theory, called Conformal Cyclic Cosmology (CCC), proposes the universe goes through infinite cycles, where the end of one "aeon" (a universe's lifespan) smoothly transitions into the Big Bang of the next. Art: Andrea Belloni, Water flowing | anbello.eth | anbello.tez, Used with permission

83,060 просмотров • 6 месяцев назад

Mathematics. Our complicated, seemingly unpredictable universe. This is chaos theory in action. The motion of a "quadruple pendulum" (simulation with no friction). Source 1 (Reddit u/tmanchester): Source 2 (Matlab code):

Mathematics. Our complicated, seemingly unpredictable universe. This is chaos theory in action. The motion of a "quadruple pendulum" (simulation with no friction). Source 1 (Reddit u/tmanchester): Source 2 (Matlab code):

381,272 просмотров • 3 лет назад

Mathematics, physics, chaos. Apparent unpredictability. Life explained. The "Lorenz Attractor." Little blue butterflies, circulating forever. Source: CC BY-NC-SA 3.0

Mathematics, physics, chaos. Apparent unpredictability. Life explained. The "Lorenz Attractor." Little blue butterflies, circulating forever. Source: CC BY-NC-SA 3.0

22,309 просмотров • 1 месяц назад

Mathematics. Physics. Magnetic Chaotic Pendulum: adding magnetic forces to gravity gives this pendulum chaotic motion (sensitivity to slight differences in initial launch conditions). By Physicsfun, Used with permission.

Mathematics. Physics. Magnetic Chaotic Pendulum: adding magnetic forces to gravity gives this pendulum chaotic motion (sensitivity to slight differences in initial launch conditions). By Physicsfun, Used with permission.

333,418 просмотров • 3 лет назад

Mathematics. Hidden inside every square is a 3-4-5 right triangle, which humans can reveal in the following manner. By math educator Tim Brzezinski, Tim Brzezinski, Used with permission.

Mathematics. Hidden inside every square is a 3-4-5 right triangle, which humans can reveal in the following manner. By math educator Tim Brzezinski, Tim Brzezinski, Used with permission.

70,753 просмотров • 6 месяцев назад

Mathematics, topology, geometry, simulation, algorithms. Watch this Cosmic Eversion Loop. Particle Tube Metamorphosis. Notice how the interior surface of the tube evaginates to become the "exterior." By Acid Boy 💊, Used with permission.

Mathematics, topology, geometry, simulation, algorithms. Watch this Cosmic Eversion Loop. Particle Tube Metamorphosis. Notice how the interior surface of the tube evaginates to become the "exterior." By Acid Boy 💊, Used with permission.

56,277 просмотров • 7 месяцев назад

Math. This mystery space-filling curve is a variant of Hilbert's curve, built from epicycles. Watch it evolve as more harmonics are added. Source: I created this animation using Hopperpop's Epicycle Drawer, an interactive tool:

Math. This mystery space-filling curve is a variant of Hilbert's curve, built from epicycles. Watch it evolve as more harmonics are added. Source: I created this animation using Hopperpop's Epicycle Drawer, an interactive tool:

31,036 просмотров • 4 месяцев назад

The year is 2020. This infographic animation caught my eye today. Covid-19. Source: CC BY-NC-SA 4.0

The year is 2020. This infographic animation caught my eye today. Covid-19. Source: CC BY-NC-SA 4.0

38,169 просмотров • 5 месяцев назад

→ Math, physics, reality. Imagine showing this animation to scientists 200 years ago, and accelerating the growth of knowledge. Credit: I didn’t upload this video to 𝕏. I’m pointing to a video residing in the 𝕏 stream of “raz_akram” (Mar 17, 2025).

→ Math, physics, reality. Imagine showing this animation to scientists 200 years ago, and accelerating the growth of knowledge. Credit: I didn’t upload this video to 𝕏. I’m pointing to a video residing in the 𝕏 stream of “raz_akram” (Mar 17, 2025).

14,468 просмотров • 1 месяц назад

Mathematics, astronomy, physics. Dance of Earth and Mercury. By Engaging Data, Used with permission.

Mathematics, astronomy, physics. Dance of Earth and Mercury. By Engaging Data, Used with permission.

42,168 просмотров • 7 месяцев назад

Mathematics, Physics. "Dirac's Belt Trick is a clever way to remove two full twists in a ribbon by simply moving it around. This gives a way of connecting cables to a rotating object without them twisting or tangling." By Matt Henderson, Matt Henderson, used with permission.

Mathematics, Physics. "Dirac's Belt Trick is a clever way to remove two full twists in a ribbon by simply moving it around. This gives a way of connecting cables to a rotating object without them twisting or tangling." By Matt Henderson, Matt Henderson, used with permission.

50,251 просмотров • 9 месяцев назад