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	2022-08-17: What If Dark Energy is a New Quantum Field? 08:36: ... while another fifth element called quintessence filled the celestial spheres beyond the earth. It’s an apt because it's field fills all of space and ...
	2022-05-04: Space DOES NOT Expand Everywhere 02:38: ... part of that geometry could be the 3-D analog of our surface of a sphere, which would be a space that loops back on itself in all directions. Or ... 05:19: ... it's embedded in a larger FLRW spacetime? Does the underlying expanding sphere fight against the inward pull of gravity? Is the space inside, say, a ... 07:12: ... of space so the universe becomes an expanding ring instead of a ,sphere, with grid points instead of gridlines. But then we can add the dimension ... 07:58: ... look like? If the galaxy causes gridlines to be pulled together on a sphere, then on the ring it causes points to be pulled together, and they ... 05:19: ... it's embedded in a larger FLRW spacetime? Does the underlying expanding sphere fight against the inward pull of gravity? Is the space inside, say, a galaxy ...
	2022-03-23: Where Is The Center of The Universe? 04:51: First imagine the surface of a sphere. 05:20: If I were to ask a denizen of the surface of the sphere to point to the center of the universe, they couldn’t do it. 05:28: They can’t point “down” to the center of the sphere, because to them there is no down. 07:39: ... line loops around the closed universe as the sphere shrinks, until eventually all points in the universe, including the ... 09:52: In the case of the closed universe that’s easier to imagine - rewind the growing sphere and it approaches a single point at t=0. 07:39: ... line loops around the closed universe as the sphere shrinks, until eventually all points in the universe, including the pointed line, ...
	2022-03-08: Is the Proxima System Our Best Hope For Another Earth? 00:02: ... was mapping what he thought to  be pin pricks in the celestial sphere. ...
	2022-02-10: The Nature of Space and Time AMA 00:03: ... really is stretched okay in that there's um if you were to define a sphere around the massive object that sphere would contain more state it ...
	2022-01-12: How To Simulate The Universe With DFT 14:43: ... the Earth, and then the one about how we might search for Dyson spheres or other alien ... 17:48: Ok, moving on to searching for Dyson spheres. 17:51: Krkrunner speculates that if a civilization is advanced enough to create a Dyson sphere they might also be able to mask it from detection. 18:10: ... Dyson sphere reprocessing a sun’s visible light into infrared, ultimately releasing ... 18:21: ... weren’t releasing as much energy as it absorbsd then the interior of the sphere would have to heat up until the input-output equilibrium is ... 18:51: ... civilization that would need the amount of energy generated by a Dyson sphere. ... 19:27: ... Dyson sphere wouldn’t actually cut it But actually, even nearly-light-speed travel ... 20:07: ... the other thing we might do with a matryoshka brain powered by a a Dyson Sphere. Which somehow seems, horrifyingly more ... 20:19: ... bitcoin mining AIs take over and cannibalize the world to build Dyson Spheres to power a hash cashing matryoshka brain. Our awesome space-faring, ... 18:10: ... Dyson sphere reprocessing a sun’s visible light into infrared, ultimately releasing exactly as ... 14:43: ... the Earth, and then the one about how we might search for Dyson spheres or other alien ... 17:48: Ok, moving on to searching for Dyson spheres. 19:27: ... nearly-light-speed travel can be a real gas guzzler, so perhaps Dyson spheres are made to power light sail lasers that accelerate craft to ... 20:19: ... bitcoin mining AIs take over and cannibalize the world to build Dyson Spheres to power a hash cashing matryoshka brain. Our awesome space-faring, ...
	2021-12-29: How to Find ALIEN Dyson Spheres 01:00: ... a lot of room to expand if we were to cover the entire sphere surrounding the Sun with the radius of Earth’s orbit, we’d collect all ... 01:27: It was also named the Stapledon sphere - but only by the guy who was trying to avoid having his own name attached to the idea, without success. 01:38: ... that a highly advanced civilization might expand to occupy an entire sphere surrounding its home star at the radius of its planet of origin, for the ... 01:56: To Dyson’s regret, the notion became called the Dyson sphere. 02:01: ... we’ve talked about the plausibility of Dyson spheres in the past - long story short, not plausible as a solid shell, but ... 02:23: ... the Dyson sphere - or swarm or whatever - is a natural stage in the progression of many ... 04:06: So how do you go about searching for a Dyson sphere? 04:29: ... added some details to the calculations, figuring out how close a Dyson sphere would need to be in order to be detected with the technology of the ... 05:20: It turns out it’s near impossible to identify a full Dyson sphere from its thermal emission alone. 05:27: ... suggested that we consider these infrared sources as candidate Dyson spheres, to be followed up by scanning for narrow-band radio emission, a much ... 05:48: Dyson and Sagan etc’s calculations were for a full Dyson sphere - one that reprocesses all of their star’s light. 06:03: Partial Dyson spheres could take the form of equatorial rings of collectors, or swarms that don’t fill the entire spherical surface. 06:18: It turns out that a partial Dyson sphere might be even more detectable than a full one. 07:02: For example, it might be surrounded by a Dyson sphere. 07:11: It would be an unnatural combination of a large, hot object - the central star, and a gigantic cool object - the partial sphere. 07:24: But from our great distance the light from the star and the sphere would be blended into a point. 07:52: But that’s a lot of work even for a single star, and we probably need to search a huge number of stars to hope to find even a single Dyson sphere. 09:14: Let’s see what a Dyson sphere would actually do to a star on the HR diagram. 09:46: ... Shiro Nishimura, Japanese astronomers who’ve been searching for Dyson spheres for years, calculate that a partial sphere that intercepts only 1% of ... 10:56: ... other approach to finding partial Dyson spheres is to look for strange variations in brightness and colour over time, as ... 11:30: Yup, still no Dyson spheres. 11:50: But that doesn’t mean Dyson spheres aren’t out there. 12:12: ... we can't identify a single Dyson sphere's effects around individual stars at distant galaxies. But what if a ... 13:11: In the past couple of years researchers have been thinking about Dyson spheres built around the supermassive black holes in the centers of galaxies. 13:19: And about the effect on the enclosed star due to a partial Dyson sphere. 01:27: It was also named the Stapledon sphere - but only by the guy who was trying to avoid having his own name attached to the idea, without success. 02:23: ... the Dyson sphere - or swarm or whatever - is a natural stage in the progression of many ... 05:48: Dyson and Sagan etc’s calculations were for a full Dyson sphere - one that reprocesses all of their star’s light. 01:00: ... a lot of room to expand if we were to cover the entire sphere surrounding the Sun with the radius of Earth’s orbit, we’d collect all of the Sun’s ... 01:38: ... that a highly advanced civilization might expand to occupy an entire sphere surrounding its home star at the radius of its planet of origin, for the purpose of ... 02:01: ... we’ve talked about the plausibility of Dyson spheres in the past - long story short, not plausible as a solid shell, but ... 05:27: ... suggested that we consider these infrared sources as candidate Dyson spheres, to be followed up by scanning for narrow-band radio emission, a much ... 06:03: Partial Dyson spheres could take the form of equatorial rings of collectors, or swarms that don’t fill the entire spherical surface. 09:46: ... Shiro Nishimura, Japanese astronomers who’ve been searching for Dyson spheres for years, calculate that a partial sphere that intercepts only 1% of ... 10:56: ... other approach to finding partial Dyson spheres is to look for strange variations in brightness and colour over time, as ... 11:30: Yup, still no Dyson spheres. 11:50: But that doesn’t mean Dyson spheres aren’t out there. 12:12: ... we can't identify a single Dyson sphere's effects around individual stars at distant galaxies. But what if a ... 13:11: In the past couple of years researchers have been thinking about Dyson spheres built around the supermassive black holes in the centers of galaxies. 12:12: ... we can't identify a single Dyson sphere's effects around individual stars at distant galaxies. But what if a civilization ...
	2021-07-21: How Magnetism Shapes The Universe 05:11: Although it’s less of a sphere and more of a teardrop - dragged into that shape by the Sun’s orbital motion through the galaxy.
	2021-05-11: How To Know If It's Aliens 17:14: ... to say is that the hypothesis is not talked about much in the public sphere, hence the ...
	2021-04-13: What If Dark Matter Is Just Black Holes? 07:58: ... other dark compact mass like a brown dwarf star, neutron star, Dyson sphere or a cluster of Reaper capital ships - as long as they’re massive, ...
	2020-12-08: Why Do You Remember The Past But Not The Future? 01:59: ... going to ignore all of the subtlety and assume the brain is a perfect sphere. ...
	2020-09-01: How Do We Know What Stars Are Made Of? 00:02: Pin-pricks in the celestial sphere, through which shines the light of heaven?
	2020-08-17: How Stars Destroy Each Other 13:42: ... other, connect, before coming together into an ovoid and then finally a sphere, or flattened sphere for a rotating black ...
	2020-05-27: Does Gravity Require Extra Dimensions? 02:22: The arrows get spread out over the surface of a sphere with the gravitating body in the center. 02:29: So the density and the strength of the gravitational field drop proportional the surface area of that sphere - that’s 4 pi r^2.
	2020-05-11: How Luminiferous Aether Led to Relativity 15:32: ... farther out into the universe, we are looking at the inner surfaces of spheres which are progressively smaller. Right - because we’re looking back in ... 16:14: ... now 93 billion light years across - so the CMB sphere is around 85 million light years across. That’s about half again the ... 15:32: ... farther out into the universe, we are looking at the inner surfaces of spheres which are progressively smaller. Right - because we’re looking back in ...
	2020-04-22: Will Wormholes Allow Fast Interstellar Travel? 03:45: ... back the 3rd spatial dimension. The ring of the wormhole is actually a sphere the size of the Schwarzschild event horizon. If we could pass through ...
	2020-01-20: Solving the Three Body Problem 09:25: ... we map that information on the surface of a sphere. We only need the 2-D surface because if we know 2 internal angles of the ...
	2019-12-02: Is The Universe Finite? 03:08: ... 2-D analogy for this is the surface of a sphere, so a our 3-D space would be like the surface of a 4-dimensional sphere - ... 03:26: Such a universe has a finite volume, just as a sphere has a finite surface area. 03:08: ... sphere, so a our 3-D space would be like the surface of a 4-dimensional sphere - also known as a ...
	2019-10-07: Black Hole Harmonics 03:02: ... harmonics of a vibrating sphere – be it a blob of water in zero-g or a black hole – are analogous to the ... 04:05: ... a set of functions pretty analogous to 2-D sine wave on the surface of a sphere, and each spherical harmonic can represent a single, pure oscillation on ...
	2019-09-23: Is Pluto a Planet? 02:07: ... 1543 Nicolaus Copernicus published “On the Revolutions of the Heavenly Spheres” which cast the Sun, not the Earth, as the center of the universe and the ... 02:27: Newton’s Laws of Gravity led to consistent predictability of the motion of the heavenly spheres. 02:07: ... 1543 Nicolaus Copernicus published “On the Revolutions of the Heavenly Spheres” which cast the Sun, not the Earth, as the center of the universe and the ... 02:27: Newton’s Laws of Gravity led to consistent predictability of the motion of the heavenly spheres.
	2019-08-12: Exploring Arecibo in VR 180 03:24: ... right now Okay, hold on to something and look straight down that giant sphere is the Gregorian dome It's like a six storey high building that contains ...
	2019-07-25: Deciphering The Vast Scale of the Universe 02:54: ... image a dome above our heads that the ancients called the celestial sphere. ...
	2019-05-16: The Cosmic Dark Ages 07:19: ... holes with millions, even billions of times the Sun’s mass – inescapable spheres the size of solar systems. And in the final stage of this feeding ...
	2019-05-09: Why Quantum Computing Requires Quantum Cryptography 16:09: What was imaged was light escaping from the photon sphere that was produced a the magnetically driven jet.
	2019-05-01: The Real Science of the EHT Black Hole 06:17: What we’re seeing here is the black hole shadow inside the bright ring of the photon sphere. 06:24: The photon sphere is where gravity is so strong that light itself can orbit the black hole. 06:36: We only see light that escapes directly towards us, so the photon sphere looks like a photon ring. 06:42: For a non-rotating black hole the sphere should be at 1.5 times the Schwarzschild radius. 06:53: Measuring the radius of the photon sphere potentially gives you both black hole mass and spin. 06:58: There are two main sources of light feeding the photon sphere. 08:03: So that synchrotron light shines from the jet vortex, it gets trapped briefly in the photon sphere, and then makes its way to us. 08:14: The ring is the photon sphere, blurred due to the fact that this observation is incredibly difficult and the EHT is “only” the size of the Earth. 08:38: ... us – or at least forwards before the light gets deflected in the photon sphere. ... 08:14: The ring is the photon sphere, blurred due to the fact that this observation is incredibly difficult and the EHT is “only” the size of the Earth.
	2019-04-24: No Dark Matter = Proof of Dark Matter? 00:03: ... stored on the surface of a volume idea had got me thinking pick a sphere that encompasses whatever you define of as me so everything about me can ...
	2019-04-03: The Edge of an Infinite Universe 02:35: ... that curves back on itself – like a 3-D analog of the 2-D surface of a sphere. ... 11:27: ... infinite ways it can be tessellated with regular polygons, while spheres and flat space each have only a small finite number of possible ...
	2019-03-06: The Impossibility of Perpetual Motion Machines 15:33: In fact it's pretty much exactly that, but for the surface of a sphere.
	2019-02-20: Secrets of the Cosmic Microwave Background 02:05: ... of like sine waves but different wavelengths but on a 2D surface of a sphere Fluctuations in each of these layers oscillate independently But by ... 14:23: ... out that the baryon-acoustic oscillations should have produced bubbles, spheres rather than rings on the sky That's right, they definitely would have ... 02:05: ... of like sine waves but different wavelengths but on a 2D surface of a sphere Fluctuations in each of these layers oscillate independently But by adding them ... 14:23: ... out that the baryon-acoustic oscillations should have produced bubbles, spheres rather than rings on the sky That's right, they definitely would have ...
	2018-10-25: Will We Ever Find Alien Life? 05:00: Let's call such a solar fleet a Dyson swarm after the Dyson sphere, which is the next level up, a structure that completely encases a star.
	2018-10-03: How to Detect Extra Dimensions 03:29: Then the same pulse would spread out over an expanding circle, not a sphere.
	2018-08-15: Quantum Theory's Most Incredible Prediction 05:09: ... you get almost exactly twice the value you'd expect for a tiny classical sphere with the same charge and angular momentum as an ...
	2018-06-13: What Survives Inside A Black Hole? 06:37: ... comes from the fact that if you draw increasingly large spheres around a pointlike field source, the intensity of those forces gets ... 06:48: You can think of lines of force getting further and further apart for larger spheres. 06:52: The area of that sphere is proportional to the square of its radius, so the density of the lines of force decrease with distance squared. 06:37: ... comes from the fact that if you draw increasingly large spheres around a pointlike field source, the intensity of those forces gets ... 06:48: You can think of lines of force getting further and further apart for larger spheres.
	2018-05-16: Noether's Theorem and The Symmetries of Reality 02:53: A perfect sphere is continuously symmetric under rotational translations.
	2018-05-02: The Star at the End of Time 09:41: ... matter exists in a puffy sphere some 200,000 light years in radius surrounding the Milky Way, compared ...
	2018-04-25: Black Hole Swarms 10:22: There may be more efficient ways to gather energy at that scale, like a good old fashioned Dyson sphere.
	2017-09-13: Neutron Stars Collide in New LIGO Signal? 02:14: These insane objects carry the mass of a star within a sphere the size of a city, around 18 kilometers in diameter.
	2017-01-04: How to See Black Holes + Kugelblitz Challenge Answer 05:25: One-- Project Phoenix Egg is to build a giant Dyson sphere just outside the moon's orbit to absorb the incoming radiation. 06:21: So one of those vertical lines represents the surface of a sphere of a given radius. 08:15: We generate a perfectly reflective sphere at approximately half the radius of the eventual event horizon. 09:11: Our other plan was to build a Dyson sphere just outside the moon's orbit. 09:23: The Dyson sphere absorbs all of the energy from the shell, so it immediately gains the entire mass equivalence, 100,000 suns worth. 09:33: Fortunately, I said that the sphere was infinitely strong. 09:39: Just above the sphere, which is only a bit larger than that event horizon that was going to form, the spacetime curvature is pretty insane. 09:48: But it's not quite a black hole, and so in principle, the sphere doesn't have to collapse. 09:53: ... that the original mass of the sphere isn't going to add enough to the 100,000 suns of the near Kugelblitz to ... 10:21: Now, some of you may still be wondering, what about the inside of the sphere? 10:36: In Einsteinian terms, spacetime is flat within the sphere. 10:41: ... blocked out the sun, but hey, we just built an infinitely strong Dyson sphere and charged it with an impossible amount of ... 09:23: The Dyson sphere absorbs all of the energy from the shell, so it immediately gains the entire mass equivalence, 100,000 suns worth. 09:48: But it's not quite a black hole, and so in principle, the sphere doesn't have to collapse. 09:53: ... that the original mass of the sphere isn't going to add enough to the 100,000 suns of the near Kugelblitz to ...
	2016-12-14: Escape The Kugelblitz Challenge 04:31: Their spaceships form a sphere around the solar system and blast a pulse of light inwards. 05:55: Plan A is to build an infinitely-strong Dyson sphere surrounding the earth just outside the moon's orbital radius. 07:12: Operation Phoenix Egg's Dyson Sphere or Operation Disco Ball's Reflective Shield? 05:55: Plan A is to build an infinitely-strong Dyson sphere surrounding the earth just outside the moon's orbital radius.
	2016-11-16: Strange Stars 12:04: Your comparison was for people living in a 2D flatland universe that has the geometry of the surface of a 3D sphere. 12:37: That arc occurs in an imaginary fourth dimension that is analogous to the radial dimension of the 3D sphere.
	2016-11-02: Quantum Vortices and Superconductivity + Drake Equation Challenge Answers 07:21: ... we instead need to ask, "How large a sphere centered on the sun would also contain 100,000 stars?" There should be, ... 07:36: Again, using the "solstation" website, we get that there are around 5,000 stars total per 100 light year radius sphere. 07:46: So you'd get 100,000 stars in a 270 light year sphere. 07:21: ... we instead need to ask, "How large a sphere centered on the sun would also contain 100,000 stars?" There should be, on ...
	2016-09-14: Self-Replicating Robots and Galactic Domination 12:23: Shawn Tripp imagines how cool would be if we started exploring the Kuiper Belt, only to find out that it is the remnants of an ancient Dyson sphere.
	2016-08-24: Should We Build a Dyson Sphere? 00:02: MATTHEW O'DOWD: This episode is supported by "The Great Courses plus." The idea of Dyson spheres has captured our imaginations. 00:27: Today we ask, are Dyson spheres plausible? 00:53: ... Dyson spheres would be possible targets for our search for extraterrestrial ... 01:35: Are Dyson spheres the way to go? 01:38: The plausibility of a solid sphere the size of a planetary orbit is not really in question. 02:09: The sphere would not be habitable, having only a tiny gravitational pull at its surface, and that would be towards the sun. 02:36: It's not feasible to build a giant solar sphere. 02:49: ... of building a Dyson sphere, build a Dyson swarm, individual solar collectors that are only ... 00:02: MATTHEW O'DOWD: This episode is supported by "The Great Courses plus." The idea of Dyson spheres has captured our imaginations. 00:27: Today we ask, are Dyson spheres plausible? 00:53: ... Dyson spheres would be possible targets for our search for extraterrestrial ... 01:35: Are Dyson spheres the way to go? 00:27: Today we ask, are Dyson spheres plausible? 08:30: The Dyson sphere/swarm can absorb at most the entire energy output of the sun.
	2016-07-06: Juno to Reveal Jupiter's Violent Past 13:07: As a physicist, I usually approximate everything as a sphere, including planets, so approximating a tortoise as a turtle is actually pretty good.
	2016-06-15: The Strange Universe of Gravitational Lensing 08:23: The photon sphere hovers at about half, again, the height of the event horizon. 08:46: ... outspiraling light escapes the photon sphere, it joins with severely lensed light from any surrounding whirlpool of ... 08:23: The photon sphere hovers at about half, again, the height of the event horizon.
	2016-04-20: Why the Universe Needs Dark Energy 03:12: Spatial snapshots of the cosmos at an instant in time will be curved like the surface of a sphere-- except the 3D surface of a 4D hypersphere. 03:30: And circles and spheres contain more internal area/volume than their circumferences or surfaces should allow.
	2016-03-23: How Cosmic Inflation Flattened the Universe 01:51: Draw the same triangle on the surface of a sphere, which has what we call positive curvature, and the angles add up to more than 180. 10:15: ... Schneider asks, "Why does dark matter in a galaxy seem to form a sphere?" Well, this is because dark matter doesn't really interact with itself ... 13:11: ... spheroid-- while a non-rotating Schwarzschild black hole is a perfect sphere. ...
	2016-03-16: Why is the Earth Round and the Milky Way Flat? 00:05: The earth is clearly a sphere and yet the Milky Way is a disk. 00:23: It loves building spheres like stars, planets, and moons, and disks like spiral galaxies, solar systems, and some crazy stuff like quasars. 02:34: ... nothing else around, a surface of constant gravitational field is a sphere. ... 03:12: The Earth is definitely a sphere, or pretty close to. 04:33: But then, why do these towers need to form a sphere? 05:12: Now, in the case of a sphere, that's cool. 06:09: That is, until it becomes a sphere. 06:39: So a relatively solid, rocky planet will fracture and reshape itself into a sphere as long as its own gravitational field is strong enough. 06:50: It turns out that any body larger than several kilometers in diameter will form a sphere. 08:34: So why don't they form spheres? 09:58: Spheres happen when pressure is the dominant effect resisting gravity. 00:23: It loves building spheres like stars, planets, and moons, and disks like spiral galaxies, solar systems, and some crazy stuff like quasars. 08:34: So why don't they form spheres? 09:58: Spheres happen when pressure is the dominant effect resisting gravity.
	2015-11-25: 100 Years of Relativity + Challenge Winners! 06:03: We'll just calculate that angle assuming Apophis is a 325-meter diameter sphere.
	2015-10-22: Have Gravitational Waves Been Discovered?!? 02:00: So a rotating sphere or a cylinder doesn't make waves.
	2015-09-30: What Happens At The Edge Of The Universe? 06:28: If that curvature is positive, then it may be that the universe is really the surface of a hypersphere, the 3D surface of a 4D sphere.
	2015-08-19: Do Events Inside Black Holes Happen? 04:52: For simplicity of presentation, let's pretend that the Sun is a perfect sphere.
	2015-08-12: Challenge: Which Particle Wins This Race? 00:32: Pretend you have a sphere with the same mass density throughout. 00:36: That sphere is not rotating, and it's not orbiting any other larger bodies. 00:40: For simplicity, I'm going to refer to this sphere as a planet, but it could be any other massive body. 02:26: ... the fact that the density is uniform, plus some basic geometry about spheres, you should be able to get a formula for the gravitational force on the ...
	2015-07-29: General Relativity & Curved Spacetime Explained! 02:21: ... using our old friend the two dimensional ant on the surface of the sphere. ... 02:35: ... an actual plane, and decides to draw an xy grid on a large patch of the sphere, with its x-axis along the equator and the y-axis along longitude ... 02:55: You can't put a big, rectangular grid on a sphere without bunching it up. 03:01: Stated another way, a sphere can accommodate local Euclidean grids in tiny patches, but not global ones. 04:32: ... that, since spacetime is curved, can and do cross just like on the sphere. ...
	2015-07-15: Can You Trust Your Eyes in Spacetime? 00:28: In the episode on curvature, we saw that tiny patches on the surface of a sphere look like the Euclidean plane. 00:33: Since we know how to draw straight line segments in a plane, then we know how to do that in tiny patches on the sphere. 00:38: So draw tiny straight segments over a series of tiny patches, join them, and voila, you have a geodesic on the sphere.
	2015-07-08: Curvature Demonstrated + Comments 00:19: The question I saw most from you guys in the comments had to do with exactly what tangency and parallel transport of vectors mean on the sphere. 00:42: Suppose I have a vector on the sphere. 00:44: Clearly that vector doesn't lie on the surface of the sphere. 00:48: Instead, it's tangent to the sphere. 00:50: Every vector we ever dealt with in this episode was tangent to the sphere. 00:54: But being tangent to the sphere at this point does not mean that you're tangent to this black curve at this point. 01:44: Now imagine zooming in on this little patch of the sphere. 01:48: ... you zoom in on that little patch of the sphere, that little patch of the sphere is barely distinguishable from an entire ... 02:03: ... the idea is that in a tiny little patch of the sphere, the ant can pretend that he's instead living on the plane that's tangent ... 02:37: ... smaller and smaller and smaller patches, drawing the curve out on the sphere that he generates through this ... 02:52: In the limit, he will end up producing this great circle, and whatever curve you generate in that limit is what we mean by a geodesic on the sphere. 04:19: ... how the dimensionality of the ambient space in which the surface of the sphere is embedded-- namely, the 3D ambient space-- affects the discussion of ... 04:31: The ambient space is just there as a visual aid for you, but everything we've been discussing is intrinsic to the sphere itself. 06:18: The fact that they do converge eventually is another indication that the surface of a sphere is curved.
	2015-07-02: Can a Circle Be a Straight Line? 02:33: ... if we want to do geometry on arbitrary spaces like on the surface of a sphere or a saddle or on some funky hillside, that's not a vacuous ... 03:32: For example, picture an ant confined to the surface of an ordinary sphere with no concept of or access to the direction off the surface. 04:18: Over tiny regions of the sphere the ant can pretend that it's on a plane, and it can use planar definitions of parallel and tangent. 05:36: Intuitively, we know a plane is flat and that a sphere is curved.
	2015-02-25: How Do You Measure the Size of the Universe? 01:01: Well, it's a sphere with a radius of about 46 billion light years, or, rounding down to keep it simple, about 90 billion light years end to end.

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