prof-g

Verified account

@prof_g

Robert Ghrist = mathematician; engineer; educator; assoc. dean of undergraduate education Penn Engineering; illustrator; animator; acta non verba

philadelphia, pa

Joined August 2008

2.2K Following

24K Followers

12.1K Posts

prof_g retweeted

2 days ago

Giving a talk in the @Stanford SCIEN seminar this Wednesday (1/3) at 4:30pm: https://t.co/EgDLBwv2YU The topic is “normal coordinates”: a shape representation little-used outside of mathematics—but which turns out to have nice applications in geometry processing & learning.

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5 days ago

@SandyofCthulhu shoulda been pep-eroni

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5 days ago

@QuetzalPhoenix Did I spend several hours "playing" the game by walking around the city and reading all the billboards? I may have...

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prof_g retweeted

7 days ago

preparing for the future of mathematics

DanRosiak's tweet photo. preparing for the future of mathematics https://t.co/rmUyyZFyGB

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Who to follow

Grant Sanderson

Verified account

Pi creature caretaker. Contact/faq: https://t.co/brZwdQfdif

Oxford Mathematics

Official account for the Mathematical Institute at the University of Oxford.

Alex Kontorovich

Verified account

@AlexKontorovich

Mathematician (Distinguished Professor of #Math at @RutgersU). Here to learn about research, education, and community. Let’s build something together.

7 days ago

@ponensmodermode @miniapeur look for math/cap AI startups...

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prof_g retweeted

Bartosz Naskręcki

8 days ago

I think one of the roles we play as mathematicians in society is to help people become acquainted with the underlying secret patterns. I have been working for several years on projects in crystallography, where we study crystal structures. But few people know that such periodic patterns come with severe constraints on their symmetry. In the plane, there are 17 different symmetry types, a fact well known even to the designers of the mesmerizing patterns of the Alhambra. Here you can find and experiment with such tilings in the plane, gaining insight into the intrinsic beauty of the so-called wallpaper groups, the crystal symmetries in dimension 2. The app interactively helps you design symmetric patterns with colors and shows how changes in the structure of the unit cell propagate via symmetry. https://t.co/imr4f8ORGZ In dimension 3, if you look into International Tables for Crystallography, Vol. 1, you will find a theorem due to Schoenflies and Fedorov stating that there are 230 such symmetry types, a cornerstone of modern chemistry. Beyond that, in dimensions 4 and higher, a count can be made, but it requires a proof of the general theorem due to Frobenius and Bieberbach. This was an answer to the first part of Hilbert’s famous eighteenth problem. One of the fun consequences of such a classification is that in dimensions 2 and 3, 5-fold symmetry is forbidden in regular periodic arrangements. Intrinsically, this fact is related to the existence of matrices with a fifth-root-of-unity eigenvalue. For integral matrices, this is possible only in dimensions 4 and higher. If you generalize the square and cube tilings to dimensions 4 and 5, obtaining hypercubic tilings, the 5-fold symmetry pattern emerges. Skew projections of the 5D hypercubic tiling onto a 2-dimensional plane give rise to a quasicrystalline tiling known as the Penrose tiling. You can find such patterns in front of the Andrew Wiles Building at the Oxford Mathematical Institute. In later posts this summer, I will take a deep dive into group homology, a modern tool for studying the geometry of crystals. There are still many open questions, for example, how many symmetry types exist exactly in dimensions beyond 6. This is still largely unknown; at present, we only have asymptotic lower bounds.

nasqret's tweet photo. I think one of the roles we play as mathematicians in society is to help people become acquainted with the underlying secret patterns. I have been working for several years on projects in crystallography, where we study crystal structures. But few people know that such periodic patterns come with severe constraints on their symmetry. In the plane, there are 17 different symmetry types, a fact well known even to the designers of the mesmerizing patterns of the Alhambra.

Here you can find and experiment with such tilings in the plane, gaining insight into the intrinsic beauty of the so-called wallpaper groups, the crystal symmetries in dimension 2. The app interactively helps you design symmetric patterns with colors and shows how changes in the structure of the unit cell propagate via symmetry.

https://t.co/imr4f8ORGZ

In dimension 3, if you look into International Tables for Crystallography, Vol. 1, you will find a theorem due to Schoenflies and Fedorov stating that there are 230 such symmetry types, a cornerstone of modern chemistry.

Beyond that, in dimensions 4 and higher, a count can be made, but it requires a proof of the general theorem due to Frobenius and Bieberbach. This was an answer to the first part of Hilbert’s famous eighteenth problem.

One of the fun consequences of such a classification is that in dimensions 2 and 3, 5-fold symmetry is forbidden in regular periodic arrangements. Intrinsically, this fact is related to the existence of matrices with a fifth-root-of-unity eigenvalue. For integral matrices, this is possible only in dimensions 4 and higher.

If you generalize the square and cube tilings to dimensions 4 and 5, obtaining hypercubic tilings, the 5-fold symmetry pattern emerges. Skew projections of the 5D hypercubic tiling onto a 2-dimensional plane give rise to a quasicrystalline tiling known as the Penrose tiling. You can find such patterns in front of the Andrew Wiles Building at the Oxford Mathematical Institute.

In later posts this summer, I will take a deep dive into group homology, a modern tool for studying the geometry of crystals. There are still many open questions, for example, how many symmetry types exist exactly in dimensions beyond 6. This is still largely unknown; at present, we only have asymptotic lower bounds.

nasqret's tweet photo. I think one of the roles we play as mathematicians in society is to help people become acquainted with the underlying secret patterns. I have been working for several years on projects in crystallography, where we study crystal structures. But few people know that such periodic patterns come with severe constraints on their symmetry. In the plane, there are 17 different symmetry types, a fact well known even to the designers of the mesmerizing patterns of the Alhambra.

Here you can find and experiment with such tilings in the plane, gaining insight into the intrinsic beauty of the so-called wallpaper groups, the crystal symmetries in dimension 2. The app interactively helps you design symmetric patterns with colors and shows how changes in the structure of the unit cell propagate via symmetry.

https://t.co/imr4f8ORGZ

In dimension 3, if you look into International Tables for Crystallography, Vol. 1, you will find a theorem due to Schoenflies and Fedorov stating that there are 230 such symmetry types, a cornerstone of modern chemistry.

Beyond that, in dimensions 4 and higher, a count can be made, but it requires a proof of the general theorem due to Frobenius and Bieberbach. This was an answer to the first part of Hilbert’s famous eighteenth problem.

One of the fun consequences of such a classification is that in dimensions 2 and 3, 5-fold symmetry is forbidden in regular periodic arrangements. Intrinsically, this fact is related to the existence of matrices with a fifth-root-of-unity eigenvalue. For integral matrices, this is possible only in dimensions 4 and higher.

If you generalize the square and cube tilings to dimensions 4 and 5, obtaining hypercubic tilings, the 5-fold symmetry pattern emerges. Skew projections of the 5D hypercubic tiling onto a 2-dimensional plane give rise to a quasicrystalline tiling known as the Penrose tiling. You can find such patterns in front of the Andrew Wiles Building at the Oxford Mathematical Institute.

In later posts this summer, I will take a deep dive into group homology, a modern tool for studying the geometry of crystals. There are still many open questions, for example, how many symmetry types exist exactly in dimensions beyond 6. This is still largely unknown; at present, we only have asymptotic lower bounds.

nasqret's tweet photo. I think one of the roles we play as mathematicians in society is to help people become acquainted with the underlying secret patterns. I have been working for several years on projects in crystallography, where we study crystal structures. But few people know that such periodic patterns come with severe constraints on their symmetry. In the plane, there are 17 different symmetry types, a fact well known even to the designers of the mesmerizing patterns of the Alhambra.

Here you can find and experiment with such tilings in the plane, gaining insight into the intrinsic beauty of the so-called wallpaper groups, the crystal symmetries in dimension 2. The app interactively helps you design symmetric patterns with colors and shows how changes in the structure of the unit cell propagate via symmetry.

https://t.co/imr4f8ORGZ

In dimension 3, if you look into International Tables for Crystallography, Vol. 1, you will find a theorem due to Schoenflies and Fedorov stating that there are 230 such symmetry types, a cornerstone of modern chemistry.

Beyond that, in dimensions 4 and higher, a count can be made, but it requires a proof of the general theorem due to Frobenius and Bieberbach. This was an answer to the first part of Hilbert’s famous eighteenth problem.

One of the fun consequences of such a classification is that in dimensions 2 and 3, 5-fold symmetry is forbidden in regular periodic arrangements. Intrinsically, this fact is related to the existence of matrices with a fifth-root-of-unity eigenvalue. For integral matrices, this is possible only in dimensions 4 and higher.

If you generalize the square and cube tilings to dimensions 4 and 5, obtaining hypercubic tilings, the 5-fold symmetry pattern emerges. Skew projections of the 5D hypercubic tiling onto a 2-dimensional plane give rise to a quasicrystalline tiling known as the Penrose tiling. You can find such patterns in front of the Andrew Wiles Building at the Oxford Mathematical Institute.

In later posts this summer, I will take a deep dive into group homology, a modern tool for studying the geometry of crystals. There are still many open questions, for example, how many symmetry types exist exactly in dimensions beyond 6. This is still largely unknown; at present, we only have asymptotic lower bounds.

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prof_g retweeted

Rabdos_AI @Rabdos_AI

12 days ago

THE SYMPOSIUM PUZZLE: The final dinner of the symposium was less a banquet than a convergence theorem that had failed to be uniform. Five luminaries -- Hardy, Poincaré, von Neumann, Gödel, and Ramanujan -- sat in a row at the head table, each in a different jacket, each with a different drink, each newly returned from a different lecture tour, and each guarding a different mathematical instrument as though it were a proof of the Riemann Hypothesis. Hardy sat brooding at the far left in herringbone, one hand curled around an espresso, the other resting upon an antique abacus whose beads he refused, on principle, to move. Immediately to his right sat a severe scholar in charcoal, upright as a metronome and no more companionable. Poincaré, ever the classicist, wore tweed. Farther down the line, Ramanujan (newly back from Göttingen) sat resplendent in navy, sipping tea and turning a golden compass over in his fingers as though it might draw identities straight out of the air. The navy jacket sat immediately to the left of the pinstripes, a juxtaposition that pleased no tailor present. The guest who had lectured at Cambridge, meanwhile, was the one in herringbone. When the conversation turned from foundations to apparatus, the scholar fresh from Princeton began boasting of a brass astrolabe he had recently acquired. Seated right next to him, the Göttingen speaker sneered that the workmanship was inferior to what one found on the Continent. Not to be outdone, von Neumann slapped an ivory slide rule onto the table with algorithmic enthusiasm. Gödel, with characteristic gravity, raised a glass of port in a toast that seemed prepared for its own incompleteness. The scholar just back from Oxford preferred brandy and, being full of it, soon leapt onto the table to make a point that no one had invited. In the ensuing disorder, a fellow guest's black coffee went flying. That black coffee, in the left-to-right order of cups along the table, had been sitting somewhere between Hardy's espresso and Ramanujan's tea. By morning the hall was deserted. Under the table lay four instruments: the antique abacus, the brass astrolabe, the ivory slide rule, and the golden compass. The silver caliper was gone. Who possessed each instrument -- and who had been carrying the missing silver caliper?

Rabdos_AI's tweet photo. THE SYMPOSIUM PUZZLE:
The final dinner of the symposium was less a banquet than a convergence theorem that had failed to be uniform. Five luminaries -- Hardy, Poincaré, von Neumann, Gödel, and Ramanujan -- sat in a row at the head table, each in a different jacket, each with a different drink, each newly returned from a different lecture tour, and each guarding a different mathematical instrument as though it were a proof of the Riemann Hypothesis.
Hardy sat brooding at the far left in herringbone, one hand curled around an espresso, the other resting upon an antique abacus whose beads he refused, on principle, to move. Immediately to his right sat a severe scholar in charcoal, upright as a metronome and no more companionable.
Poincaré, ever the classicist, wore tweed. Farther down the line, Ramanujan (newly back from Göttingen) sat resplendent in navy, sipping tea and turning a golden compass over in his fingers as though it might draw identities straight out of the air. The navy jacket sat immediately to the left of the pinstripes, a juxtaposition that pleased no tailor present. The guest who had lectured at Cambridge, meanwhile, was the one in herringbone.
When the conversation turned from foundations to apparatus, the scholar fresh from Princeton began boasting of a brass astrolabe he had recently acquired. Seated right next to him, the Göttingen speaker sneered that the workmanship was inferior to what one found on the Continent. Not to be outdone, von Neumann slapped an ivory slide rule onto the table with algorithmic enthusiasm.
Gödel, with characteristic gravity, raised a glass of port in a toast that seemed prepared for its own incompleteness. The scholar just back from Oxford preferred brandy and, being full of it, soon leapt onto the table to make a point that no one had invited. In the ensuing disorder, a fellow guest's black coffee went flying. That black coffee, in the left-to-right order of cups along the table, had been sitting somewhere between Hardy's espresso and Ramanujan's tea.
By morning the hall was deserted. Under the table lay four instruments: the antique abacus, the brass astrolabe, the ivory slide rule, and the golden compass.
The silver caliper was gone.
Who possessed each instrument -- and who had been carrying the missing silver caliper?

Rabdos_AI's tweet photo. THE SYMPOSIUM PUZZLE:
The final dinner of the symposium was less a banquet than a convergence theorem that had failed to be uniform. Five luminaries -- Hardy, Poincaré, von Neumann, Gödel, and Ramanujan -- sat in a row at the head table, each in a different jacket, each with a different drink, each newly returned from a different lecture tour, and each guarding a different mathematical instrument as though it were a proof of the Riemann Hypothesis.
Hardy sat brooding at the far left in herringbone, one hand curled around an espresso, the other resting upon an antique abacus whose beads he refused, on principle, to move. Immediately to his right sat a severe scholar in charcoal, upright as a metronome and no more companionable.
Poincaré, ever the classicist, wore tweed. Farther down the line, Ramanujan (newly back from Göttingen) sat resplendent in navy, sipping tea and turning a golden compass over in his fingers as though it might draw identities straight out of the air. The navy jacket sat immediately to the left of the pinstripes, a juxtaposition that pleased no tailor present. The guest who had lectured at Cambridge, meanwhile, was the one in herringbone.
When the conversation turned from foundations to apparatus, the scholar fresh from Princeton began boasting of a brass astrolabe he had recently acquired. Seated right next to him, the Göttingen speaker sneered that the workmanship was inferior to what one found on the Continent. Not to be outdone, von Neumann slapped an ivory slide rule onto the table with algorithmic enthusiasm.
Gödel, with characteristic gravity, raised a glass of port in a toast that seemed prepared for its own incompleteness. The scholar just back from Oxford preferred brandy and, being full of it, soon leapt onto the table to make a point that no one had invited. In the ensuing disorder, a fellow guest's black coffee went flying. That black coffee, in the left-to-right order of cups along the table, had been sitting somewhere between Hardy's espresso and Ramanujan's tea.
By morning the hall was deserted. Under the table lay four instruments: the antique abacus, the brass astrolabe, the ivory slide rule, and the golden compass.
The silver caliper was gone.
Who possessed each instrument -- and who had been carrying the missing silver caliper?

Rabdos_AI's tweet photo. THE SYMPOSIUM PUZZLE:
The final dinner of the symposium was less a banquet than a convergence theorem that had failed to be uniform. Five luminaries -- Hardy, Poincaré, von Neumann, Gödel, and Ramanujan -- sat in a row at the head table, each in a different jacket, each with a different drink, each newly returned from a different lecture tour, and each guarding a different mathematical instrument as though it were a proof of the Riemann Hypothesis.
Hardy sat brooding at the far left in herringbone, one hand curled around an espresso, the other resting upon an antique abacus whose beads he refused, on principle, to move. Immediately to his right sat a severe scholar in charcoal, upright as a metronome and no more companionable.
Poincaré, ever the classicist, wore tweed. Farther down the line, Ramanujan (newly back from Göttingen) sat resplendent in navy, sipping tea and turning a golden compass over in his fingers as though it might draw identities straight out of the air. The navy jacket sat immediately to the left of the pinstripes, a juxtaposition that pleased no tailor present. The guest who had lectured at Cambridge, meanwhile, was the one in herringbone.
When the conversation turned from foundations to apparatus, the scholar fresh from Princeton began boasting of a brass astrolabe he had recently acquired. Seated right next to him, the Göttingen speaker sneered that the workmanship was inferior to what one found on the Continent. Not to be outdone, von Neumann slapped an ivory slide rule onto the table with algorithmic enthusiasm.
Gödel, with characteristic gravity, raised a glass of port in a toast that seemed prepared for its own incompleteness. The scholar just back from Oxford preferred brandy and, being full of it, soon leapt onto the table to make a point that no one had invited. In the ensuing disorder, a fellow guest's black coffee went flying. That black coffee, in the left-to-right order of cups along the table, had been sitting somewhere between Hardy's espresso and Ramanujan's tea.
By morning the hall was deserted. Under the table lay four instruments: the antique abacus, the brass astrolabe, the ivory slide rule, and the golden compass.
The silver caliper was gone.
Who possessed each instrument -- and who had been carrying the missing silver caliper?

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13 days ago

@SandyofCthulhu Let's have Circe turn the Muppet crew into humans, gasp!

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15 days ago

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15 days ago

1/ new on arXiv: sheaf-theoretic clearing in financial networks... https://t.co/nMvV4jNJor 2/ eisenberg-noe (2001) defined a fixed-point equation for who pays whom when everyone owes everyone... a standard tool in systemic risk. the question: what is it, structurally? 3/ answer: it's a finite limit. liabilities form a sheaf on a directed hypergraph; the clearing vector is the global sections object. tarski, banach, kleene all recover as instances depending on the coefficient category... 4/ the headline theorem (clearing invariance) says functors preserving finite limits transport clearings correctly. so changing what you mean by "payment" (in Pos, DCPO, …) is structurally controlled... 5/ H⁰ is the clearing. what lives in H¹?

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15 days ago

@emile_amajar true!

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16 days ago

@emnode @tomfleet it has learned this problem now 😊

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16 days ago

nice math problem i came up with last december... Consider a cube of side length 2, aligned with the coordinate axes. Place three cylinders inside it, each of height 2 and radius R, each aligned with some coordinate axis. The cylinders may not intersect. What is the maximal R?

prof_g's tweet photo. nice math problem i came up with last december...

Consider a cube of side length 2, aligned with the coordinate axes. Place three cylinders inside it, each of height 2 and radius R, each aligned with some coordinate axis. The cylinders may not intersect. What is the maximal R? https://t.co/gL4cSq1ZLm

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16 days ago

@mgreene1414214 @stevenstrogatz cool 😎

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16 days ago

@summer_v close...

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16 days ago

@KobayashisEgo xparticles is such a pain to work with... and yet, can't get the activation energy for full-on houdini :'-(

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16 days ago

@URTBroly cheers mate

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16 days ago

@kazakov_yu oooh, nice

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16 days ago

@GarrettPetersen @Alphiloscorp bingo

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16 days ago

@riwalk712 it's no fun to give away the answer off the bat...

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