Mastodawn

Another post on the #cosmicdistanceLadder instagram of Tanya Klowden and myself, this time comparing the many ways cartographers have projected the (mostly) spherical Earth onto flat planes. No planar projection can faithfully reproduce *all* the geometric features of a sphere, so each projection is a compromise; but some projections are still preferred over others for specific applications. https://www.instagram.com/p/DVC9SvxkXDs

Cosmic Distance Ladder on Instagram: "The Earth is round, but maps (or computer and phone screens) are flat. So, we have to apply a projection to map the round Earth onto a flat surface. Ideally, we would like this projection to have three desirable properties: • Area-preserving. The projection should not stretch one part of the Earth to look larger than another part that actually has the same surface area. • Shape-preserving.(Mathematicians call this the “angle-preserving” or “conformal” property.) The projection should not squish or stretch the shape of an object: a round landmass should not become elliptical, for instance. • Grid lines. Lines of latitude should be horizontal; and lines of longitude (also known as “meridians”) should be vertical. Unfortunately, these three properties form a trilemma: any given projection can have two of these three properties, but not all three at once. The Mercator projection is the most famous. It is shape-preserving and has grid lines, making it particularly suitable for navigation. But it distorts area like crazy, making Greenland for instance look almost three times larger than Australia by area, when in fact it is over three times smaller. The Gall-Peters projection removes the area distortion, and keeps the grid lines - but at the cost of totally messing up shapes. Africa is way too tall and skinny; Canada too short and wide. The Mollweide projection tries its best to preserve both area and shape, and gives up entirely on straight grid lines. The relationship between area and shape is in fact fundamentally different for spheres and for planes, so no projection can preserve both of them perfectly; but Mollweide comes the closest. Can’t choose? Why not try the Winkel Tripel projection, which compromises by being only slightly bad at preserving areas and shapes, or keeping the grid-lines straight. Everything in this projection is slightly imperfect, but it tries to balance its faults evenly. There are an infinite number of possible projections. What projection is your favorite? #DistanceLadder #geography #MapProjections #3DGeometry #topology #mathematics #WhyNobodyLikesMercatorProjections"

27 likes, 1 comments - cosmic_distance_ladder on February 21, 2026: "The Earth is round, but maps (or computer and phone screens) are flat. So, we have to apply a projection to map the round Earth onto a flat surface. Ideally, we would like this projection to have three desirable properties: • Area-preserving. The projection should not stretch one part of the Earth to look larger than another part that actually has the same surface area. • Shape-preserving.(Mathematicians call this the “angle-preserving” or “conformal” property.) The projection should not squish or stretch the shape of an object: a round landmass should not become elliptical, for instance. • Grid lines. Lines of latitude should be horizontal; and lines of longitude (also known as “meridians”) should be vertical. Unfortunately, these three properties form a trilemma: any given projection can have two of these three properties, but not all three at once. The Mercator projection is the most famous. It is shape-preserving and has grid lines, making it particularly suitable for navigation. But it distorts area like crazy, making Greenland for instance look almost three times larger than Australia by area, when in fact it is over three times smaller. The Gall-Peters projection removes the area distortion, and keeps the grid lines - but at the cost of totally messing up shapes. Africa is way too tall and skinny; Canada too short and wide. The Mollweide projection tries its best to preserve both area and shape, and gives up entirely on straight grid lines. The relationship between area and shape is in fact fundamentally different for spheres and for planes, so no projection can preserve both of them perfectly; but Mollweide comes the closest. Can’t choose? Why not try the Winkel Tripel projection, which compromises by being only slightly bad at preserving areas and shapes, or keeping the grid-lines straight. Everything in this projection is slightly imperfect, but it tries to balance its faults evenly. There are an infinite number of possible projections. What projection is your favorite? #DistanceLadder #geography #MapProjections #3DGeometry #topology #mathematics #WhyNobodyLikesMercatorProjections".

Instagram

Terence Tao Jun 20, 2025

A new #CosmicDistanceLadder post to mark the summer solstice, on how astronomical measurements, from the time of Eratosthenes to the modern day, rely on the tireless (and often unsung) efforts of many careful and precise data collectors. https://www.instagram.com/p/DLG6a_WIWyb

Cosmic Distance Ladder on Instagram: "In the fourth century BCE, Eratosthenes took advantage of the small deviation between the angle of the summer solstice Sun in the cities of Syene (more or less on the Tropic of Cancer) and Alexandria (hundreds of miles to the north) to obtain the first reasonably accurate, recorded measurement of the circumference of the Earth. For this, he needed to know the precise distance between the two cities. We have joked that Eratosthenes hired a “graduate student” to walk between the towns. The truth may not be too far off—in that era there were professional bematists whose job was to precisely measure long distances, either through methodical pacing or with early odometers. The history of the cosmic distance ladder is replete with such carefully collected data which has always relied on two things: well-calibrated tools and a plethora of tireless researchers whose names have frequently been lost to history. We see this in the many scholars of Tycho Brahe’s Uraniborg collecting decades of heroic naked-eye astronomical observations. It is also in the adept organization and remarkable spectroscopic analysis of the glass plate photographs at the Harvard Observatory by dozens of the Harvard “Computers”—skilled women who developed the ability to categorize the spectra of hundreds of stars an hour and maintained a working knowledge of the features of thousands of bright stars in the northern and southern hemispheres. Today, teams of hundreds of scientists collect many terabytes of observational data streamed to us from a wide range of complex telescopes on Earth and delicately calibrated specialized instruments in space observatories. Eratosthenes would likely be baffled by the technology and sophisticated mathematics of modern astronomy, but perhaps he would recognize that it is still fundamentally the same type of science, driven by both data and theoretical reasoning, that it was in his day. He would certainly recognize the importance of the countless unsung researchers who wrangle the volumes of data that astronomy demands. #DistanceLadder #astronomy #bematist #DataScience #AncientAstronomy #RenaissanceAstronomy #HarvardObservatory #VeraCRubinObservatory"

3 likes, 0 comments - cosmic_distance_ladder on June 19, 2025: "In the fourth century BCE, Eratosthenes took advantage of the small deviation between the angle of the summer solstice Sun in the cities of Syene (more or less on the Tropic of Cancer) and Alexandria (hundreds of miles to the north) to obtain the first reasonably accurate, recorded measurement of the circumference of the Earth. For this, he needed to know the precise distance between the two cities. We have joked that Eratosthenes hired a “graduate student” to walk between the towns. The truth may not be too far off—in that era there were professional bematists whose job was to precisely measure long distances, either through methodical pacing or with early odometers. The history of the cosmic distance ladder is replete with such carefully collected data which has always relied on two things: well-calibrated tools and a plethora of tireless researchers whose names have frequently been lost to history. We see this in the many scholars of Tycho Brahe’s Uraniborg collecting decades of heroic naked-eye astronomical observations. It is also in the adept organization and remarkable spectroscopic analysis of the glass plate photographs at the Harvard Observatory by dozens of the Harvard “Computers”—skilled women who developed the ability to categorize the spectra of hundreds of stars an hour and maintained a working knowledge of the features of thousands of bright stars in the northern and southern hemispheres. Today, teams of hundreds of scientists collect many terabytes of observational data streamed to us from a wide range of complex telescopes on Earth and delicately calibrated specialized instruments in space observatories. Eratosthenes would likely be baffled by the technology and sophisticated mathematics of modern astronomy, but perhaps he would recognize that it is still fundamentally the same type of science, driven by both data and theoretical reasoning, that it was in his day. He would certainly recognize the importance of the countless unsung researchers who wrangle the volumes of data that astronomy demands. #DistanceLadder #astronomy #bematist #DataScience #AncientAstronomy #RenaissanceAstronomy #HarvardObservatory #VeraCRubinObservatory".

Instagram

Terence Tao May 19, 2025

A new #CosmicDistanceLadder post on how the distance ladder can also be used to measure cosmic durations, as well as cosmic distances. https://www.instagram.com/p/DJ2ra2soG4j

Cosmic Distance Ladder on Instagram: "The cosmic distance ladder, as the name suggests, is used to measure the vast distances in the universe. But it also can be used to measure the vast durations of time in the universe as well, providing answers to such questions as "How old is the Earth?", "How old is the Sun?", and "How old is the universe?". These measurements, when combined with techniques from the more terrestrial sciences of geochronology, history, and archeology (such as Carbon-14 dating and tree ring counting), have given us a consistent timeline of our planet and our universe spanning billions of years. But the story was not always so harmonious. For instance, when Hubble first discovered his eponymous law on the expanding nature of the universe in the 1920s, his calculations suggested that the universe could only have existed for about 1.8 billion years. Meanwhile, using the recently discovered phenomenon of radioactive decay, early geochronologists such as Bertram Boltwood and Arthur Holmes dated the crystallization times of minerals containing radioactive isotopes such as Uranium-238 to be approximately 2-3 billion years. So, for a time, the best scientific estimates were predicting the Earth to be older than the entire universe! Fortunately, this issue was resolved with more precise measurements, as well as finer theoretical models of cosmology, nuclear physics, and isotope diffusion. Modern radiometric dating places the age of the Earth at about 4.5 billion years, while the age of the universe is widely believed to be about 13.8 billion years. Some discrepancies still remain, however. There is an ongoing "Hubble tension" between two ways of measuring the constant in Hubble's law—one from direct redshift measurements of distant objects in the universe, and the other by inferring the constant from fitting cosmological models to observed fluctuations in the cosmic microwave background. These measurements are now precise to within 1-2%, yet differ by 10% from each other. The jury is still out on exactly how this tension will be resolved, but it is hardly the first "crisis in cosmology"... #DistanceLadder #astronomy #TimeLadder #cosmology #Hubble #CrisisInCosmology"

1 likes, 0 comments - cosmic_distance_ladder on May 19, 2025: "The cosmic distance ladder, as the name suggests, is used to measure the vast distances in the universe. But it also can be used to measure the vast durations of time in the universe as well, providing answers to such questions as "How old is the Earth?", "How old is the Sun?", and "How old is the universe?". These measurements, when combined with techniques from the more terrestrial sciences of geochronology, history, and archeology (such as Carbon-14 dating and tree ring counting), have given us a consistent timeline of our planet and our universe spanning billions of years. But the story was not always so harmonious. For instance, when Hubble first discovered his eponymous law on the expanding nature of the universe in the 1920s, his calculations suggested that the universe could only have existed for about 1.8 billion years. Meanwhile, using the recently discovered phenomenon of radioactive decay, early geochronologists such as Bertram Boltwood and Arthur Holmes dated the crystallization times of minerals containing radioactive isotopes such as Uranium-238 to be approximately 2-3 billion years. So, for a time, the best scientific estimates were predicting the Earth to be older than the entire universe! Fortunately, this issue was resolved with more precise measurements, as well as finer theoretical models of cosmology, nuclear physics, and isotope diffusion. Modern radiometric dating places the age of the Earth at about 4.5 billion years, while the age of the universe is widely believed to be about 13.8 billion years. Some discrepancies still remain, however. There is an ongoing "Hubble tension" between two ways of measuring the constant in Hubble's law—one from direct redshift measurements of distant objects in the universe, and the other by inferring the constant from fitting cosmological models to observed fluctuations in the cosmic microwave background. These measurements are now precise to within 1-2%, yet differ by 10% from each other. The jury is still out on exactly how this tension will be resolved, but it is hardly the first "crisis in cosmology"... #DistanceLadder #astronomy #TimeLadder #cosmology #Hubble #CrisisInCosmology".

Instagram

Terence Tao Apr 10, 2025

A new #CosmicDistanceLadder post, on intriguing hints from the DESI survey data that suggests that the cosmological constant (aka "dark energy) might not, in fact, be constant after all. https://www.instagram.com/p/DIP0yy5oDUu

Cosmic Distance Ladder on Instagram: "When Einstein developed his general theory of relativity, he noticed that his equations predicted the universe could expand and then contract, or expand forever; but without additional counteracting laws of physics, it could not stay at a steady size indefinitely. In order to make his theory better fit the prevailing “steady state” model of the universe of his time, Einstein added a term to his equations, involving an unknown “cosmological constant” Λ (Lambda), that caused the vacuum of space itself to inherently expand and counteract the contracting force of gravity. Although evidence from the 1930s onwards favored an expanding universe (the “Big Bang” model) and Einstein removed this constant from his equations, reportedly calling it his “biggest blunder”, we now view this constant as one of the simplest forms of “dark energy”. But in the 1990s, two large research teams, headed by Saul Perlmitter, and Adam Riess and Brian P. Schmidt, respectively, precisely measured the distance from the Earth of supernovae designated type Ia (“one-A”). Both projects independently discovered that the expansion of the universe was not slowing (as they expected) but accelerating over time. The simplest way to explain these facts was to reinstate a (small) cosmological constant Λ, leading to the standard “Λ-CDM” model of the universe today (the CDM stands for “cold dark matter”). Perlmutter, Riess, and Schmidt received the 2011 Nobel Prize in Physics for the discovery. Now, new analysis from an ongoing survey using the Dark Energy Spectroscopic Instrument (DESI) suggests this acceleration is itself fluctuating in time. It seems dark energy, usually treated as synonymous with the “cosmological constant” is not actually constant after all, but potentially evolving. As the study confirms the existence of the mysterious phenomenon of dark energy, it simultaneously shows that there is much about it that we still don’t understand. The last rung in the Cosmic Distance Ladder is “How big is our universe?”. For now, each new result seems to suggest it is growing bigger than any previous model believed it could be. #DistanceLadder #astronomy #DarkEnergy #astrophysics #cosmology #DESI"

19 likes, 1 comments - cosmic_distance_ladder on April 9, 2025: "When Einstein developed his general theory of relativity, he noticed that his equations predicted the universe could expand and then contract, or expand forever; but without additional counteracting laws of physics, it could not stay at a steady size indefinitely. In order to make his theory better fit the prevailing “steady state” model of the universe of his time, Einstein added a term to his equations, involving an unknown “cosmological constant” Λ (Lambda), that caused the vacuum of space itself to inherently expand and counteract the contracting force of gravity. Although evidence from the 1930s onwards favored an expanding universe (the “Big Bang” model) and Einstein removed this constant from his equations, reportedly calling it his “biggest blunder”, we now view this constant as one of the simplest forms of “dark energy”. But in the 1990s, two large research teams, headed by Saul Perlmitter, and Adam Riess and Brian P. Schmidt, respectively, precisely measured the distance from the Earth of supernovae designated type Ia (“one-A”). Both projects independently discovered that the expansion of the universe was not slowing (as they expected) but accelerating over time. The simplest way to explain these facts was to reinstate a (small) cosmological constant Λ, leading to the standard “Λ-CDM” model of the universe today (the CDM stands for “cold dark matter”). Perlmutter, Riess, and Schmidt received the 2011 Nobel Prize in Physics for the discovery. Now, new analysis from an ongoing survey using the Dark Energy Spectroscopic Instrument (DESI) suggests this acceleration is itself fluctuating in time. It seems dark energy, usually treated as synonymous with the “cosmological constant” is not actually constant after all, but potentially evolving. As the study confirms the existence of the mysterious phenomenon of dark energy, it simultaneously shows that there is much about it that we still don’t understand. The last rung in the Cosmic Distance Ladder is “How big is our universe?”. For now, each new result seems to suggest it is growing bigger than any previous model believed it could be. #DistanceLadder #astronomy #DarkEnergy #astrophysics #cosmology #DESI".

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Level 98 🇺🇦Mar 24, 2025

@tao #Physics Terrence Tao has put some great posts and links (to stuff / videos etc. he's done with Grant etc.) on the hashtag #CosmicDistanceLadder. I encourage anyone interested in physics, astro etc. to have a look through these if they haven't already.

Terence Tao Mar 24, 2025

A new #CosmicDistanceLadder post on why lunar and solar eclipses tend to come in pairs (for instance, the solar eclipse next week is paired with the lunar eclipse from last week). https://www.instagram.com/p/DHkS3EcA40L

Cosmic Distance Ladder on Instagram: "Anaxagoras (c. 500 BCE) is one of the earliest natural philosophers we know of to propose what is essentially the modern explanation of eclipses, which is that they are caused when the positions of the Earth, Moon, and Sun line up in such a way that the shadow of the Earth falls on the Moon (a lunar eclipse), or the shadow of the Moon falls on the Earth (a solar eclipse). The ancient Greeks also gave us the term for this type of alignment: syzygy (which means ‘paired together’). The Ancient Greek model explains other features of eclipses with relatively few additional assumptions. For instance, one can observe that lunar eclipses happen during what would otherwise be a full Moon, and solar eclipses happen during what would otherwise be a new Moon. Similarly, once one accepts the additional axiom that the Moon orbits the Earth in a roughly circular (but perhaps inclined) orbit once every lunar month (about four weeks), one can also explain the observed phenomenon that lunar and solar eclipses tend to come in pairs that are half a lunar month (or two weeks) apart; for instance, the partial solar eclipse occurring on Saturday, March 29th (this weekend!) is paired with the total lunar eclipse from last week. Indeed, this is the time needed for the Moon to move from a syzygy of opposition to the Sun to one of conjunction, or vice versa. Aristarchus cleverly used both lunar and solar eclipses, as well as the half-Moon phases, to determine the size and distances of both the Moon and Sun. We’ll go into this story in great detail in the book, but in the meantime you can check out the video with @3blue1brown on this at: https://youtu.be/YdOXS_9_P4U?t=488 (and in the Linktree in our bio!) #DistanceLadder #astronomy #LunarEclipse #SolarEclipse #syzygy #AncientAstronomy #GreekAstronomy #mythology #TotalLunarEclipse2025 #PartialSolarEclipse2025"

13 likes, 0 comments - cosmic_distance_ladder on March 23, 2025: "Anaxagoras (c. 500 BCE) is one of the earliest natural philosophers we know of to propose what is essentially the modern explanation of eclipses, which is that they are caused when the positions of the Earth, Moon, and Sun line up in such a way that the shadow of the Earth falls on the Moon (a lunar eclipse), or the shadow of the Moon falls on the Earth (a solar eclipse). The ancient Greeks also gave us the term for this type of alignment: syzygy (which means ‘paired together’). The Ancient Greek model explains other features of eclipses with relatively few additional assumptions. For instance, one can observe that lunar eclipses happen during what would otherwise be a full Moon, and solar eclipses happen during what would otherwise be a new Moon. Similarly, once one accepts the additional axiom that the Moon orbits the Earth in a roughly circular (but perhaps inclined) orbit once every lunar month (about four weeks), one can also explain the observed phenomenon that lunar and solar eclipses tend to come in pairs that are half a lunar month (or two weeks) apart; for instance, the partial solar eclipse occurring on Saturday, March 29th (this weekend!) is paired with the total lunar eclipse from last week. Indeed, this is the time needed for the Moon to move from a syzygy of opposition to the Sun to one of conjunction, or vice versa. Aristarchus cleverly used both lunar and solar eclipses, as well as the half-Moon phases, to determine the size and distances of both the Moon and Sun. We’ll go into this story in great detail in the book, but in the meantime you can check out the video with @3blue1brown on this at: https://youtu.be/YdOXS_9_P4U?t=488 (and in the Linktree in our bio!) #DistanceLadder #astronomy #LunarEclipse #SolarEclipse #syzygy #AncientAstronomy #GreekAstronomy #mythology #TotalLunarEclipse2025 #PartialSolarEclipse2025".

Instagram

Terence Tao Mar 17, 2025

A new #CosmicDistanceLadder post, on how the recent lunar eclipse from the vantage point of the Earth becomes a solar eclipse from the vantage point of the Moon: https://www.instagram.com/p/DHR1tuWonDR/

Cosmic Distance Ladder on Instagram: "A change in perspective can dramatically alter the way one views objects in space. Terry observes, “I have a vivid childhood memory of being in an airplane taking off during a heavy thunderstorm, only to encounter sunny blue skies once the plane rose past the rainclouds. It was surreal to see the storm continuing to rage below while I was enjoying perfect weather.” With a greater change in location, the change of perspective can be even more dramatic. On Thursday night, many parts of the world were treated to a full lunar eclipse, in which the Sun, Earth (in the middle), and Moon align in a straight line. From our perspective, the Moon enters the shadow of the Earth, first partially and then fully. Thanks to Firefly Aerospace’s Blue Ghost lunar lander, we can also view this event from the perspective of the Moon—but as a solar eclipse, not a lunar one! From the Moon’s point of view, the Sun is being blocked—partially at first and then fully—by the intervening Earth. For solar eclipses on the Earth, the Moon and Sun are almost exactly the same angular width. In a total solar eclipse, this gives a spectacular view of the Sun’s outer atmosphere, or solar corona, which is normally hidden by the exponentially greater brightness of the Sun’s surface. On the Moon, the Earth is about three times larger in angular width; but one still observes a slightly different, but equally striking “diamond ring” effect, as the atmosphere of the Earth refracts sunlight around its border. As red light has a longer wavelength and is less prone to scattering than blue light; this small amount of refracted light is enough to dimly illuminate the Moon in a red glow as seen from the Earth, even during totality. Humanity has only ever seen a lunar eclipse from the surface of the Moon three times, all captured using unmanned probes. Observations of both solar and lunar eclipses on Earth, however, were key to figuring out the distance from the Earth to the Moon and the Earth to the Sun long before we had the technology to land on the Moon’s surface and see Earth from its perspective. #DistanceLadder #astronomy #LunarEclipse #SolarEclipse #LunarOrbit #LunarLander #perspective"

23 likes, 0 comments - cosmic_distance_ladder on March 16, 2025: "A change in perspective can dramatically alter the way one views objects in space. Terry observes, “I have a vivid childhood memory of being in an airplane taking off during a heavy thunderstorm, only to encounter sunny blue skies once the plane rose past the rainclouds. It was surreal to see the storm continuing to rage below while I was enjoying perfect weather.” With a greater change in location, the change of perspective can be even more dramatic. On Thursday night, many parts of the world were treated to a full lunar eclipse, in which the Sun, Earth (in the middle), and Moon align in a straight line. From our perspective, the Moon enters the shadow of the Earth, first partially and then fully. Thanks to Firefly Aerospace’s Blue Ghost lunar lander, we can also view this event from the perspective of the Moon—but as a solar eclipse, not a lunar one! From the Moon’s point of view, the Sun is being blocked—partially at first and then fully—by the intervening Earth. For solar eclipses on the Earth, the Moon and Sun are almost exactly the same angular width. In a total solar eclipse, this gives a spectacular view of the Sun’s outer atmosphere, or solar corona, which is normally hidden by the exponentially greater brightness of the Sun’s surface. On the Moon, the Earth is about three times larger in angular width; but one still observes a slightly different, but equally striking “diamond ring” effect, as the atmosphere of the Earth refracts sunlight around its border. As red light has a longer wavelength and is less prone to scattering than blue light; this small amount of refracted light is enough to dimly illuminate the Moon in a red glow as seen from the Earth, even during totality. Humanity has only ever seen a lunar eclipse from the surface of the Moon three times, all captured using unmanned probes. Observations of both solar and lunar eclipses on Earth, however, were key to figuring out the distance from the Earth to the Moon and the Earth to the Sun long before we had the technology to land on the Moon’s surface and see Earth from its perspective. #DistanceLadder #astronomy #LunarEclipse #SolarEclipse #LunarOrbit #LunarLander #perspective".

Instagram

Terence Tao Mar 5, 2025

A new post on my #CosmicDistanceLadder Instagram with Tanya Klowden on the parallels (but also differences) between ancient Greek and ancient Indian astronomy. https://www.instagram.com/p/DGzJs02AbBA

Cosmic Distance Ladder on Instagram: "The Distance Ladder repeatedly exemplifies how science works, with results that are independently reproducible: we consistently discover a single, objective reality. For example, we can look at the ancient Indian astronomical tradition to find principles almost parallel to familiar Greek discoveries. While there was some cultural exchanges between the Greeks and Indians, starting from the conquests of Alexander the Great, Indian astronomers largely based their work on local observations and philosophical traditions, without being constrained by the orthodoxies of Greek mathematics, physics, and cosmology. As a consequence, they developed levels of mathematical sophistication and astronomical precision that eluded the Greeks. The modern sine and cosine functions can be traced back at least to the work of Aryabhata in the 5th century CE, as do the famous Hindu-Arabic numerals (including zero) that we still use today, as well as even some rudiments of modern algebra (which would be more fully developed in the later Islamic golden age). The Greeks had already proposed the rotation of Earth around its axis and that the shadows of the Earth and Moon cause lunar and solar eclipses respectively. The Indians, with their innovative trigonometry, were able to argue more convincingly for this based on their observational data—allowing for precise predictions of future eclipses. Extended observations of planetary conjunctions allowed the Indians to compile astronomical tables that would remain accurate for thousands or even millions of years, which - many centuries later - indirectly helped Copernicus calculate the periods of the planets in his heliocentric model to sufficient accuracy to supply Kepler with the crucial ingredients needed to determine his famous laws. While the Greek contribution to scientific thought emphasized axiomatic reasoning and of justifying conclusions as a logical consequence of hypotheses, Indian astronomy was very sophisticated in its mathematics and built a strong scientific tradition that continues to inspire Indian (and other) astronomers today. #DistanceLadder #astronomy #IndianAstronomy #HistoryOfScience #Mathematics #Trigonometry"

27 likes, 1 comments - cosmic_distance_ladder on March 4, 2025: "The Distance Ladder repeatedly exemplifies how science works, with results that are independently reproducible: we consistently discover a single, objective reality. For example, we can look at the ancient Indian astronomical tradition to find principles almost parallel to familiar Greek discoveries. While there was some cultural exchanges between the Greeks and Indians, starting from the conquests of Alexander the Great, Indian astronomers largely based their work on local observations and philosophical traditions, without being constrained by the orthodoxies of Greek mathematics, physics, and cosmology. As a consequence, they developed levels of mathematical sophistication and astronomical precision that eluded the Greeks. The modern sine and cosine functions can be traced back at least to the work of Aryabhata in the 5th century CE, as do the famous Hindu-Arabic numerals (including zero) that we still use today, as well as even some rudiments of modern algebra (which would be more fully developed in the later Islamic golden age). The Greeks had already proposed the rotation of Earth around its axis and that the shadows of the Earth and Moon cause lunar and solar eclipses respectively. The Indians, with their innovative trigonometry, were able to argue more convincingly for this based on their observational data—allowing for precise predictions of future eclipses. Extended observations of planetary conjunctions allowed the Indians to compile astronomical tables that would remain accurate for thousands or even millions of years, which - many centuries later - indirectly helped Copernicus calculate the periods of the planets in his heliocentric model to sufficient accuracy to supply Kepler with the crucial ingredients needed to determine his famous laws. While the Greek contribution to scientific thought emphasized axiomatic reasoning and of justifying conclusions as a logical consequence of hypotheses, Indian astronomy was very sophisticated in its mathematics and built a strong scientific tradition that continues to inspire Indian (and other) astronomers today. #DistanceLadder #astronomy #IndianAstronomy #HistoryOfScience #Mathematics #Trigonometry".

Instagram

Hacker News Feb 25, 2025

Part two of Grant Sanderson's video with Terry Tao on the cosmic distance ladder — https://mathstodon.xyz/@tao/114054291471216181
#HackerNews #GrantSanderson #TerryTao #CosmicDistanceLadder #MathEducation #VideoSeries

Terence Tao (@[email protected])

The second part of Grant Sanderson's video interview with myself on the cosmic distance ladder is now out: https://www.youtube.com/watch?v=hFMaT9oRbs4 I wrote a blog post with additional commentary and corrections on both videos at https://terrytao.wordpress.com/2025/02/13/cosmic-distance-ladder-video-with-grant-sanderson-3blue1brown-commentary-and-corrections/

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