Re: [math-fun] Differential geometry in *finite* fields!
Perhaps I'm just weird, but I found Wildberger's intro far more motivating than most math presentations. I would imagine that proper motivation gets you 90% of the way home. I'm really sick to death of math presentations that squeeze all of the fun out of math. Yes, it's fun to abstract away all of the inessential details when trying to generalize a proof; but that's only after you got to the proof in the first place. Yes, the view from the top of Kilimanjaro is wonderful, but some of us want to appreciate it by climbing it, rather than simply looking at a photo. For example, I *never* had a decent presentation of Galois theory, even though it's perhaps the most taught piece of undergraduate algebra. I might have thought that after several hundred years, someone would have figured out how to teach these concepts to high schoolers, but apparently not. At 07:16 AM 8/1/2017, Andy Latto wrote:
While he refers to this as "differential geometry", it's the very beginning of a very large and well-studied field generally described as "algebraic geometry".. While I wouldn't discourage interested amateurs from further exploration, because I don't see any benefit in discouraging people from exploring any sort of math that interests them, it's a very well-studied field where it's very unlikely that interested amateurs will find something new.
Andy
On Tue, Aug 1, 2017 at 9:32 AM, Henry Baker <hbaker1@pipeline.com> wrote:
FYI --
I ran across this really cool video about how to do differential geometry in *finite* fields.
Suppose you have a polynomial in a finite field.
You can have a "tangent" line, a "tangent" conic, a "tangent" cubic, etc.
https://www.youtube.com/watch?v=2yGuKIz2wfE
DiffGeom7: Differential geometry with finite fields
njwildberger
https://www.youtube.com/channel/UCXl0Zbk8_rvjyLwAR-Xh9pQ
With an algebraic approach to differential geometry, the possibility of working over finite fields emerges. This is another key advantage to following Newton, Euler and Lagrange when it comes to calculus!
In this lecture we introduce the basics of finite (prime) fields, where we work mod p for some fixed prime p, and show that our study of tangent conics to a cubic polynomial extends naturally, and leads to interesting combinatorial structures. There are many possible directions for investigation by interested amateurs who have understood this lecture.
After the basics of arithmetic over the field F_p, including a discussion of primitive roots and Fermat's theorem, we discuss polynomial arithmetic and illustrate tangent conics to a particular cubic over F_11. In particular Ghys' lovely observation about the disjointness of such tangent conics (for a cubic) can be illustrated completely here, and some additional patterns visibly emerge from the vertices of the various tangent conics.
One big difference here is that the sub-derivatives and the derivatives are NOT equivalent in general, and we must replace the usual Taylor expansions with one involving sub-derivatives. Some remarks about the useful distinction between polynomials and polynomial functions in this setting are made.
This lecture shows that the calculus is actually a much wider operational tool than is usually appreciated----finite calculus not only makes sense but is a rich source of both combinatorial and algebraic patterns---and questions for further investigations.
Andy.Latto@pobox.com
I'm really sick to death of math presentations that squeeze all of the fun out of math.
Maybe it has already been mentioned on this list (I was dropped for a year), but there is a marvelous book that I'm reading by John Stillwell from 2016 called Elements of Mathematics: From Euclid to Goedel (Princeton). It is a kind of updated version of the old Felix Klein books, Elementary Math from an Advanced Standpoint, and it is wonderful. I'm learning a lot. For instance, on page 100 he discusses "Post's tag system". There were already three sequences in the OEIS based on it, A284116, A284119, A284121, and in the last few days I've added a bunch more, A289670-A289677. As Allan Wechsler has pointed out, the main open question is, is there a binary word which has an infinite orbit under the tag system? See A284116 for more information. If there is anyone on this list who knows about the present status of this question, please send me an email... And all these sequences could use more terms - and insight. Best regards Neil Neil J. A. Sloane, President, OEIS Foundation. 11 South Adelaide Avenue, Highland Park, NJ 08904, USA. Also Visiting Scientist, Math. Dept., Rutgers University, Piscataway, NJ. Phone: 732 828 6098; home page: http://NeilSloane.com Email: njasloane@gmail.com On Tue, Aug 1, 2017 at 10:48 AM, Henry Baker <hbaker1@pipeline.com> wrote:
Perhaps I'm just weird, but I found Wildberger's intro far more motivating than most math presentations.
I would imagine that proper motivation gets you 90% of the way home.
I'm really sick to death of math presentations that squeeze all of the fun out of math. Yes, it's fun to abstract away all of the inessential details when trying to generalize a proof; but that's only after you got to the proof in the first place.
Yes, the view from the top of Kilimanjaro is wonderful, but some of us want to appreciate it by climbing it, rather than simply looking at a photo.
For example, I *never* had a decent presentation of Galois theory, even though it's perhaps the most taught piece of undergraduate algebra. I might have thought that after several hundred years, someone would have figured out how to teach these concepts to high schoolers, but apparently not.
At 07:16 AM 8/1/2017, Andy Latto wrote:
While he refers to this as "differential geometry", it's the very beginning of a very large and well-studied field generally described as "algebraic geometry".. While I wouldn't discourage interested amateurs from further exploration, because I don't see any benefit in discouraging people from exploring any sort of math that interests them, it's a very well-studied field where it's very unlikely that interested amateurs will find something new.
Andy
On Tue, Aug 1, 2017 at 9:32 AM, Henry Baker <hbaker1@pipeline.com> wrote:
FYI --
I ran across this really cool video about how to do differential geometry in *finite* fields.
Suppose you have a polynomial in a finite field.
You can have a "tangent" line, a "tangent" conic, a "tangent" cubic, etc.
https://www.youtube.com/watch?v=2yGuKIz2wfE
DiffGeom7: Differential geometry with finite fields
njwildberger
https://www.youtube.com/channel/UCXl0Zbk8_rvjyLwAR-Xh9pQ
With an algebraic approach to differential geometry, the possibility of working over finite fields emerges. This is another key advantage to following Newton, Euler and Lagrange when it comes to calculus!
In this lecture we introduce the basics of finite (prime) fields, where we work mod p for some fixed prime p, and show that our study of tangent conics to a cubic polynomial extends naturally, and leads to interesting combinatorial structures. There are many possible directions for investigation by interested amateurs who have understood this lecture.
After the basics of arithmetic over the field F_p, including a discussion of primitive roots and Fermat's theorem, we discuss polynomial arithmetic and illustrate tangent conics to a particular cubic over F_11. In particular Ghys' lovely observation about the disjointness of such tangent conics (for a cubic) can be illustrated completely here, and some additional patterns visibly emerge from the vertices of the various tangent conics.
One big difference here is that the sub-derivatives and the derivatives are NOT equivalent in general, and we must replace the usual Taylor expansions with one involving sub-derivatives. Some remarks about the useful distinction between polynomials and polynomial functions in this setting are made.
This lecture shows that the calculus is actually a much wider operational tool than is usually appreciated----finite calculus not only makes sense but is a rich source of both combinatorial and algebraic patterns---and questions for further investigations.
Andy.Latto@pobox.com
_______________________________________________ math-fun mailing list math-fun@mailman.xmission.com https://mailman.xmission.com/cgi-bin/mailman/listinfo/math-fun
Do any of you know Norman Wildberger (the man or his work) at all well? I gather that some of his views are a bit far out. For instance, I believe he's said that we should get rid of irrational numbers! Though perhaps he's just saying that to be provocative, and his actual attitude is more nuanced. (Compare with Doron Zeilberger.) Jim Propp On Tue, Aug 1, 2017 at 1:08 PM, Neil Sloane <njasloane@gmail.com> wrote:
I'm really sick to death of math presentations that squeeze all of the fun out of math.
Maybe it has already been mentioned on this list (I was dropped for a year), but there is a marvelous book that I'm reading by John Stillwell from 2016 called Elements of Mathematics: From Euclid to Goedel (Princeton). It is a kind of updated version of the old Felix Klein books, Elementary Math from an Advanced Standpoint, and it is wonderful. I'm learning a lot.
For instance, on page 100 he discusses "Post's tag system". There were already three sequences in the OEIS based on it, A284116, A284119, A284121, and in the last few days I've added a bunch more, A289670-A289677. As Allan Wechsler has pointed out, the main open question is, is there a binary word which has an infinite orbit under the tag system? See A284116 for more information. If there is anyone on this list who knows about the present status of this question, please send me an email... And all these sequences could use more terms - and insight. Best regards Neil
Neil J. A. Sloane, President, OEIS Foundation. 11 South Adelaide Avenue, Highland Park, NJ 08904, USA. Also Visiting Scientist, Math. Dept., Rutgers University, Piscataway, NJ. Phone: 732 828 6098; home page: http://NeilSloane.com Email: njasloane@gmail.com
On Tue, Aug 1, 2017 at 10:48 AM, Henry Baker <hbaker1@pipeline.com> wrote:
Perhaps I'm just weird, but I found Wildberger's intro far more motivating than most math presentations.
I would imagine that proper motivation gets you 90% of the way home.
I'm really sick to death of math presentations that squeeze all of the fun out of math. Yes, it's fun to abstract away all of the inessential details when trying to generalize a proof; but that's only after you got to the proof in the first place.
Yes, the view from the top of Kilimanjaro is wonderful, but some of us want to appreciate it by climbing it, rather than simply looking at a photo.
For example, I *never* had a decent presentation of Galois theory, even though it's perhaps the most taught piece of undergraduate algebra. I might have thought that after several hundred years, someone would have figured out how to teach these concepts to high schoolers, but apparently not.
At 07:16 AM 8/1/2017, Andy Latto wrote:
While he refers to this as "differential geometry", it's the very beginning of a very large and well-studied field generally described as "algebraic geometry".. While I wouldn't discourage interested amateurs from further exploration, because I don't see any benefit in discouraging people from exploring any sort of math that interests them, it's a very well-studied field where it's very unlikely that interested amateurs will find something new.
Andy
On Tue, Aug 1, 2017 at 9:32 AM, Henry Baker <hbaker1@pipeline.com> wrote:
FYI --
I ran across this really cool video about how to do differential geometry in *finite* fields.
Suppose you have a polynomial in a finite field.
You can have a "tangent" line, a "tangent" conic, a "tangent" cubic, etc.
https://www.youtube.com/watch?v=2yGuKIz2wfE
DiffGeom7: Differential geometry with finite fields
njwildberger
https://www.youtube.com/channel/UCXl0Zbk8_rvjyLwAR-Xh9pQ
With an algebraic approach to differential geometry, the possibility of working over finite fields emerges. This is another key advantage to following Newton, Euler and Lagrange when it comes to calculus!
In this lecture we introduce the basics of finite (prime) fields, where we work mod p for some fixed prime p, and show that our study of tangent conics to a cubic polynomial extends naturally, and leads to interesting combinatorial structures. There are many possible directions for investigation by interested amateurs who have understood this lecture.
After the basics of arithmetic over the field F_p, including a discussion of primitive roots and Fermat's theorem, we discuss polynomial arithmetic and illustrate tangent conics to a particular cubic over F_11. In particular Ghys' lovely observation about the disjointness of such tangent conics (for a cubic) can be illustrated completely here, and some additional patterns visibly emerge from the vertices of the various tangent conics.
One big difference here is that the sub-derivatives and the derivatives are NOT equivalent in general, and we must replace the usual Taylor expansions with one involving sub-derivatives. Some remarks about the useful distinction between polynomials and polynomial functions in this setting are made.
This lecture shows that the calculus is actually a much wider operational tool than is usually appreciated----finite calculus not only makes sense but is a rich source of both combinatorial and algebraic patterns---and questions for further investigations.
Andy.Latto@pobox.com
_______________________________________________ math-fun mailing list math-fun@mailman.xmission.com https://mailman.xmission.com/cgi-bin/mailman/listinfo/math-fun
_______________________________________________ math-fun mailing list math-fun@mailman.xmission.com https://mailman.xmission.com/cgi-bin/mailman/listinfo/math-fun
I've been contacted by a few Wildberger fans in the past who seemed to be pretty far off the deep end. I haven't made it far enough through one of his videos to know if he says sensible things, but the appearance of his name raises my hackles. It's possible that his manner of talking about concepts that are pretty far-out for the layman just attracts people with an airy-fairy mindset. On Tue, 1 Aug 2017 at 18:32 James Propp <jamespropp@gmail.com> wrote:
Do any of you know Norman Wildberger (the man or his work) at all well? I gather that some of his views are a bit far out. For instance, I believe he's said that we should get rid of irrational numbers! Though perhaps he's just saying that to be provocative, and his actual attitude is more nuanced. (Compare with Doron Zeilberger.)
Jim Propp
On Tue, Aug 1, 2017 at 1:08 PM, Neil Sloane <njasloane@gmail.com> wrote:
I'm really sick to death of math presentations that squeeze all of the fun out of math.
Maybe it has already been mentioned on this list (I was dropped for a year), but there is a marvelous book that I'm reading by John Stillwell from 2016 called Elements of Mathematics: From Euclid to Goedel (Princeton). It is a kind of updated version of the old Felix Klein books, Elementary Math from an Advanced Standpoint, and it is wonderful. I'm learning a lot.
For instance, on page 100 he discusses "Post's tag system". There were already three sequences in the OEIS based on it, A284116, A284119, A284121, and in the last few days I've added a bunch more, A289670-A289677. As Allan Wechsler has pointed out, the main open question is, is there a binary word which has an infinite orbit under the tag system? See A284116 for more information. If there is anyone on this list who knows about the present status of this question, please send me an email... And all these sequences could use more terms - and insight. Best regards Neil
Neil J. A. Sloane, President, OEIS Foundation. 11 South Adelaide Avenue, Highland Park, NJ 08904, USA. Also Visiting Scientist, Math. Dept., Rutgers University, Piscataway, NJ. Phone: 732 828 6098 <(732)%20828-6098>; home page: http://NeilSloane.com Email: njasloane@gmail.com
On Tue, Aug 1, 2017 at 10:48 AM, Henry Baker <hbaker1@pipeline.com> wrote:
Perhaps I'm just weird, but I found Wildberger's intro far more motivating than most math presentations.
I would imagine that proper motivation gets you 90% of the way home.
I'm really sick to death of math presentations that squeeze all of the fun out of math. Yes, it's fun to abstract away all of the inessential details when trying to generalize a proof; but that's only after you got to the proof in the first place.
Yes, the view from the top of Kilimanjaro is wonderful, but some of us want to appreciate it by climbing it, rather than simply looking at a photo.
For example, I *never* had a decent presentation of Galois theory, even though it's perhaps the most taught piece of undergraduate algebra. I might have thought that after several hundred years, someone would have figured out how to teach these concepts to high schoolers, but apparently not.
At 07:16 AM 8/1/2017, Andy Latto wrote:
While he refers to this as "differential geometry", it's the very beginning of a very large and well-studied field generally described as "algebraic geometry".. While I wouldn't discourage interested amateurs from further exploration, because I don't see any benefit in discouraging people from exploring any sort of math that interests them, it's a very well-studied field where it's very unlikely that interested amateurs will find something new.
Andy
On Tue, Aug 1, 2017 at 9:32 AM, Henry Baker <hbaker1@pipeline.com> wrote:
FYI --
I ran across this really cool video about how to do differential geometry in *finite* fields.
Suppose you have a polynomial in a finite field.
You can have a "tangent" line, a "tangent" conic, a "tangent" cubic, etc.
https://www.youtube.com/watch?v=2yGuKIz2wfE
DiffGeom7: Differential geometry with finite fields
njwildberger
https://www.youtube.com/channel/UCXl0Zbk8_rvjyLwAR-Xh9pQ
With an algebraic approach to differential geometry, the possibility of working over finite fields emerges. This is another key advantage to following Newton, Euler and Lagrange when it comes to calculus!
In this lecture we introduce the basics of finite (prime) fields, where we work mod p for some fixed prime p, and show that our study of tangent conics to a cubic polynomial extends naturally, and leads to interesting combinatorial structures. There are many possible directions for investigation by interested amateurs who have understood this lecture.
After the basics of arithmetic over the field F_p, including a discussion of primitive roots and Fermat's theorem, we discuss polynomial arithmetic and illustrate tangent conics to a particular cubic over F_11. In particular Ghys' lovely observation about the disjointness of such tangent conics (for a cubic) can be illustrated completely here, and some additional patterns visibly emerge from the vertices of the various tangent conics.
One big difference here is that the sub-derivatives and the derivatives are NOT equivalent in general, and we must replace the usual Taylor expansions with one involving sub-derivatives. Some remarks about the useful distinction between polynomials and polynomial functions in this setting are made.
This lecture shows that the calculus is actually a much wider operational tool than is usually appreciated----finite calculus not only makes sense but is a rich source of both combinatorial and algebraic patterns---and questions for further investigations.
Andy.Latto@pobox.com
_______________________________________________ math-fun mailing list math-fun@mailman.xmission.com https://mailman.xmission.com/cgi-bin/mailman/listinfo/math-fun
_______________________________________________ math-fun mailing list math-fun@mailman.xmission.com https://mailman.xmission.com/cgi-bin/mailman/listinfo/math-fun
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Oh, here we go - a book called "DIVINE PROPORTIONS: Rational Trigonometry to Universal Geometry", which claims to replaces the normal trig functions with "rational functions and elementary arithmetic".
From the heading "revolutionary content": "It develops a complete theory of planar Euclidean geometry over a general field without any reliance on `axioms'." http://web.maths.unsw.edu.au/~norman/book.htm
On Thu, 3 Aug 2017 at 08:19 Christian Lawson-Perfect < christianperfect@gmail.com> wrote:
I've been contacted by a few Wildberger fans in the past who seemed to be pretty far off the deep end. I haven't made it far enough through one of his videos to know if he says sensible things, but the appearance of his name raises my hackles. It's possible that his manner of talking about concepts that are pretty far-out for the layman just attracts people with an airy-fairy mindset.
On Tue, 1 Aug 2017 at 18:32 James Propp <jamespropp@gmail.com> wrote:
Do any of you know Norman Wildberger (the man or his work) at all well? I gather that some of his views are a bit far out. For instance, I believe he's said that we should get rid of irrational numbers! Though perhaps he's just saying that to be provocative, and his actual attitude is more nuanced. (Compare with Doron Zeilberger.)
Jim Propp
On Tue, Aug 1, 2017 at 1:08 PM, Neil Sloane <njasloane@gmail.com> wrote:
I'm really sick to death of math presentations that squeeze all of the fun out of math.
Maybe it has already been mentioned on this list (I was dropped for a year), but there is a marvelous book that I'm reading by John Stillwell from 2016 called Elements of Mathematics: From Euclid to Goedel (Princeton). It is a kind of updated version of the old Felix Klein books, Elementary Math from an Advanced Standpoint, and it is wonderful. I'm learning a lot.
For instance, on page 100 he discusses "Post's tag system". There were already three sequences in the OEIS based on it, A284116, A284119, A284121, and in the last few days I've added a bunch more, A289670-A289677. As Allan Wechsler has pointed out, the main open question is, is there a binary word which has an infinite orbit under the tag system? See A284116 for more information. If there is anyone on this list who knows about the present status of this question, please send me an email... And all these sequences could use more terms - and insight. Best regards Neil
Neil J. A. Sloane, President, OEIS Foundation. 11 South Adelaide Avenue, Highland Park, NJ 08904, USA. Also Visiting Scientist, Math. Dept., Rutgers University, Piscataway, NJ. Phone: 732 828 6098 <(732)%20828-6098>; home page: http://NeilSloane.com Email: njasloane@gmail.com
On Tue, Aug 1, 2017 at 10:48 AM, Henry Baker <hbaker1@pipeline.com> wrote:
Perhaps I'm just weird, but I found Wildberger's intro far more motivating than most math presentations.
I would imagine that proper motivation gets you 90% of the way home.
I'm really sick to death of math presentations that squeeze all of the fun out of math. Yes, it's fun to abstract away all of the inessential details when trying to generalize a proof; but that's only after you got to the proof in the first place.
Yes, the view from the top of Kilimanjaro is wonderful, but some of us want to appreciate it by climbing it, rather than simply looking at a photo.
For example, I *never* had a decent presentation of Galois theory, even though it's perhaps the most taught piece of undergraduate algebra. I might have thought that after several hundred years, someone would have figured out how to teach these concepts to high schoolers, but apparently not.
At 07:16 AM 8/1/2017, Andy Latto wrote:
While he refers to this as "differential geometry", it's the very beginning of a very large and well-studied field generally described as "algebraic geometry".. While I wouldn't discourage interested amateurs from further exploration, because I don't see any benefit in discouraging people from exploring any sort of math that interests them, it's a very well-studied field where it's very unlikely that interested amateurs will find something new.
Andy
On Tue, Aug 1, 2017 at 9:32 AM, Henry Baker <hbaker1@pipeline.com> wrote:
FYI --
I ran across this really cool video about how to do differential geometry in *finite* fields.
Suppose you have a polynomial in a finite field.
You can have a "tangent" line, a "tangent" conic, a "tangent" cubic, etc.
https://www.youtube.com/watch?v=2yGuKIz2wfE
DiffGeom7: Differential geometry with finite fields
njwildberger
https://www.youtube.com/channel/UCXl0Zbk8_rvjyLwAR-Xh9pQ
With an algebraic approach to differential geometry, the possibility of working over finite fields emerges. This is another key advantage to following Newton, Euler and Lagrange when it comes to calculus!
In this lecture we introduce the basics of finite (prime) fields, where we work mod p for some fixed prime p, and show that our study of tangent conics to a cubic polynomial extends naturally, and leads to interesting combinatorial structures. There are many possible directions for investigation by interested amateurs who have understood this lecture.
After the basics of arithmetic over the field F_p, including a discussion of primitive roots and Fermat's theorem, we discuss polynomial arithmetic and illustrate tangent conics to a particular cubic over F_11. In particular Ghys' lovely observation about the disjointness of such tangent conics (for a cubic) can be illustrated completely here, and some additional patterns visibly emerge from the vertices of the various tangent conics.
One big difference here is that the sub-derivatives and the derivatives are NOT equivalent in general, and we must replace the usual Taylor expansions with one involving sub-derivatives. Some remarks about the useful distinction between polynomials and polynomial functions in this setting are made.
This lecture shows that the calculus is actually a much wider operational tool than is usually appreciated----finite calculus not only makes sense but is a rich source of both combinatorial and algebraic patterns---and questions for further investigations.
Andy.Latto@pobox.com
_______________________________________________ math-fun mailing list math-fun@mailman.xmission.com https://mailman.xmission.com/cgi-bin/mailman/listinfo/math-fun
_______________________________________________ math-fun mailing list math-fun@mailman.xmission.com https://mailman.xmission.com/cgi-bin/mailman/listinfo/math-fun
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I bought "Divine Proportions" when it first came out, and I thought it was pretty interesting. It seemed a serious work, full of original ideas. Best regards Neil Neil J. A. Sloane, President, OEIS Foundation. 11 South Adelaide Avenue, Highland Park, NJ 08904, USA. Also Visiting Scientist, Math. Dept., Rutgers University, Piscataway, NJ. Phone: 732 828 6098; home page: http://NeilSloane.com Email: njasloane@gmail.com On Thu, Aug 3, 2017 at 3:51 AM, Christian Lawson-Perfect <christianperfect@gmail.com> wrote:
Oh, here we go - a book called "DIVINE PROPORTIONS: Rational Trigonometry to Universal Geometry", which claims to replaces the normal trig functions with "rational functions and elementary arithmetic". From the heading "revolutionary content": "It develops a complete theory of planar Euclidean geometry over a general field without any reliance on `axioms'." http://web.maths.unsw.edu.au/~norman/book.htm
On Thu, 3 Aug 2017 at 08:19 Christian Lawson-Perfect < christianperfect@gmail.com> wrote:
I've been contacted by a few Wildberger fans in the past who seemed to be pretty far off the deep end. I haven't made it far enough through one of his videos to know if he says sensible things, but the appearance of his name raises my hackles. It's possible that his manner of talking about concepts that are pretty far-out for the layman just attracts people with an airy-fairy mindset.
On Tue, 1 Aug 2017 at 18:32 James Propp <jamespropp@gmail.com> wrote:
Do any of you know Norman Wildberger (the man or his work) at all well? I gather that some of his views are a bit far out. For instance, I believe he's said that we should get rid of irrational numbers! Though perhaps he's just saying that to be provocative, and his actual attitude is more nuanced. (Compare with Doron Zeilberger.)
Jim Propp
On Tue, Aug 1, 2017 at 1:08 PM, Neil Sloane <njasloane@gmail.com> wrote:
I'm really sick to death of math presentations that squeeze all of the fun out of math.
Maybe it has already been mentioned on this list (I was dropped for a year), but there is a marvelous book that I'm reading by John Stillwell from 2016 called Elements of Mathematics: From Euclid to Goedel (Princeton). It is a kind of updated version of the old Felix Klein books, Elementary Math from an Advanced Standpoint, and it is wonderful. I'm learning a lot.
For instance, on page 100 he discusses "Post's tag system". There were already three sequences in the OEIS based on it, A284116, A284119, A284121, and in the last few days I've added a bunch more, A289670-A289677. As Allan Wechsler has pointed out, the main open question is, is there a binary word which has an infinite orbit under the tag system? See A284116 for more information. If there is anyone on this list who knows about the present status of this question, please send me an email... And all these sequences could use more terms - and insight. Best regards Neil
Neil J. A. Sloane, President, OEIS Foundation. 11 South Adelaide Avenue, Highland Park, NJ 08904, USA. Also Visiting Scientist, Math. Dept., Rutgers University, Piscataway, NJ. Phone: 732 828 6098 <(732)%20828-6098>; home page: http://NeilSloane.com Email: njasloane@gmail.com
On Tue, Aug 1, 2017 at 10:48 AM, Henry Baker <hbaker1@pipeline.com> wrote:
Perhaps I'm just weird, but I found Wildberger's intro far more motivating than most math presentations.
I would imagine that proper motivation gets you 90% of the way home.
I'm really sick to death of math presentations that squeeze all of the fun out of math. Yes, it's fun to abstract away all of the inessential details when trying to generalize a proof; but that's only after you got to the proof in the first place.
Yes, the view from the top of Kilimanjaro is wonderful, but some of us want to appreciate it by climbing it, rather than simply looking at a photo.
For example, I *never* had a decent presentation of Galois theory, even though it's perhaps the most taught piece of undergraduate algebra. I might have thought that after several hundred years, someone would have figured out how to teach these concepts to high schoolers, but apparently not.
At 07:16 AM 8/1/2017, Andy Latto wrote:
While he refers to this as "differential geometry", it's the very beginning of a very large and well-studied field generally described as "algebraic geometry".. While I wouldn't discourage interested amateurs from further exploration, because I don't see any benefit in discouraging people from exploring any sort of math that interests them, it's a very well-studied field where it's very unlikely that interested amateurs will find something new.
Andy
On Tue, Aug 1, 2017 at 9:32 AM, Henry Baker <hbaker1@pipeline.com> wrote: > FYI -- > > I ran across this really cool video about how to do differential geometry in *finite* fields. > > Suppose you have a polynomial in a finite field. > > You can have a "tangent" line, a "tangent" conic, a "tangent" cubic, etc. > > https://www.youtube.com/watch?v=2yGuKIz2wfE > > DiffGeom7: Differential geometry with finite fields > > njwildberger > > https://www.youtube.com/channel/UCXl0Zbk8_rvjyLwAR-Xh9pQ > > With an algebraic approach to differential geometry, the possibility of working over finite fields emerges. This is another key advantage to following Newton, Euler and Lagrange when it comes to calculus! > > In this lecture we introduce the basics of finite (prime) fields, where we work mod p for some fixed prime p, and show that our study of tangent conics to a cubic polynomial extends naturally, and leads to interesting combinatorial structures. There are many possible directions for investigation by interested amateurs who have understood this lecture. > > After the basics of arithmetic over the field F_p, including a discussion of primitive roots and Fermat's theorem, we discuss polynomial arithmetic and illustrate tangent conics to a particular cubic over F_11. In particular Ghys' lovely observation about the disjointness of such tangent conics (for a cubic) can be illustrated completely here, and some additional patterns visibly emerge from the vertices of the various tangent conics. > > One big difference here is that the sub-derivatives and the derivatives are NOT equivalent in general, and we must replace the usual Taylor expansions with one involving sub-derivatives. Some remarks about the useful distinction between polynomials and polynomial functions in this setting are made. > > This lecture shows that the calculus is actually a much wider operational tool than is usually appreciated----finite calculus not only makes sense but is a rich source of both combinatorial and algebraic patterns---and questions for further investigations.
Andy.Latto@pobox.com
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Wildberger does himself no favours with the professional community, because he takes no interest in situating his self-invented methods in the broader scheme of established mathematics. I have a good deal of sympathy with his attitude, as it happens; but there does come a point where one must down tools and make an effort to get connected, or else risk wasting further time and temper re-inventing the wheel. My impression of "Divine Proportions" for example was that it went some of the way towards rediscovering Clifford algebra; but given how that has failed to gain acceptance over the past 144 years, maybe a fresh approach might be no bad thing! His exuberance and blackboard technique in the video were wondrous to behold. [ I was reminded of another supercharged eccentric, Simon Peyton Jones on Haskell programming --- https://www.youtube.com/watch?v=6COvD8oynmI --- and several more ... ] WFL On 8/3/17, Neil Sloane <njasloane@gmail.com> wrote:
I bought "Divine Proportions" when it first came out, and I thought it was pretty interesting. It seemed a serious work, full of original ideas. Best regards Neil
Neil J. A. Sloane, President, OEIS Foundation. 11 South Adelaide Avenue, Highland Park, NJ 08904, USA. Also Visiting Scientist, Math. Dept., Rutgers University, Piscataway, NJ. Phone: 732 828 6098; home page: http://NeilSloane.com Email: njasloane@gmail.com
On Thu, Aug 3, 2017 at 3:51 AM, Christian Lawson-Perfect <christianperfect@gmail.com> wrote:
Oh, here we go - a book called "DIVINE PROPORTIONS: Rational Trigonometry to Universal Geometry", which claims to replaces the normal trig functions with "rational functions and elementary arithmetic". From the heading "revolutionary content": "It develops a complete theory of planar Euclidean geometry over a general field without any reliance on `axioms'." http://web.maths.unsw.edu.au/~norman/book.htm
On Thu, 3 Aug 2017 at 08:19 Christian Lawson-Perfect < christianperfect@gmail.com> wrote:
I've been contacted by a few Wildberger fans in the past who seemed to be pretty far off the deep end. I haven't made it far enough through one of his videos to know if he says sensible things, but the appearance of his name raises my hackles. It's possible that his manner of talking about concepts that are pretty far-out for the layman just attracts people with an airy-fairy mindset.
On Tue, 1 Aug 2017 at 18:32 James Propp <jamespropp@gmail.com> wrote:
Do any of you know Norman Wildberger (the man or his work) at all well? I gather that some of his views are a bit far out. For instance, I believe he's said that we should get rid of irrational numbers! Though perhaps he's just saying that to be provocative, and his actual attitude is more nuanced. (Compare with Doron Zeilberger.)
Jim Propp
On Tue, Aug 1, 2017 at 1:08 PM, Neil Sloane <njasloane@gmail.com> wrote:
I'm really sick to death of math presentations that squeeze all of the fun out of math.
Maybe it has already been mentioned on this list (I was dropped for a year), but there is a marvelous book that I'm reading by John Stillwell from 2016 called Elements of Mathematics: From Euclid to Goedel (Princeton). It is a kind of updated version of the old Felix Klein books, Elementary Math from an Advanced Standpoint, and it is wonderful. I'm learning a lot.
For instance, on page 100 he discusses "Post's tag system". There were already three sequences in the OEIS based on it, A284116, A284119, A284121, and in the last few days I've added a bunch more, A289670-A289677. As Allan Wechsler has pointed out, the main open question is, is there a binary word which has an infinite orbit under the tag system? See A284116 for more information. If there is anyone on this list who knows about the present status of this question, please send me an email... And all these sequences could use more terms - and insight. Best regards Neil
Neil J. A. Sloane, President, OEIS Foundation. 11 South Adelaide Avenue, Highland Park, NJ 08904, USA. Also Visiting Scientist, Math. Dept., Rutgers University, Piscataway, NJ. Phone: 732 828 6098 <(732)%20828-6098>; home page: http://NeilSloane.com Email: njasloane@gmail.com
On Tue, Aug 1, 2017 at 10:48 AM, Henry Baker <hbaker1@pipeline.com> wrote:
Perhaps I'm just weird, but I found Wildberger's intro far more motivating than most math presentations.
I would imagine that proper motivation gets you 90% of the way home.
I'm really sick to death of math presentations that squeeze all of the fun out of math. Yes, it's fun to abstract away all of the inessential details when trying to generalize a proof; but that's only after you got to the proof in the first place.
Yes, the view from the top of Kilimanjaro is wonderful, but some of us want to appreciate it by climbing it, rather than simply looking at a photo.
For example, I *never* had a decent presentation of Galois theory, even though it's perhaps the most taught piece of undergraduate algebra. I might have thought that after several hundred years, someone would have figured out how to teach these concepts to high schoolers, but apparently not.
At 07:16 AM 8/1/2017, Andy Latto wrote: >While he refers to this as "differential geometry", it's the very >beginning of a very large and well-studied field generally > described >as "algebraic geometry".. While I wouldn't discourage interested >amateurs from further exploration, because I don't see any benefit > in >discouraging people from exploring any sort of math that interests >them, it's a very well-studied field where it's very unlikely that >interested amateurs will find something new. > >Andy > >On Tue, Aug 1, 2017 at 9:32 AM, Henry Baker <hbaker1@pipeline.com> wrote: >> FYI -- >> >> I ran across this really cool video about how to do differential geometry in *finite* fields. >> >> Suppose you have a polynomial in a finite field. >> >> You can have a "tangent" line, a "tangent" conic, a "tangent" >> cubic, etc. >> >> https://www.youtube.com/watch?v=2yGuKIz2wfE >> >> DiffGeom7: Differential geometry with finite fields >> >> njwildberger >> >> https://www.youtube.com/channel/UCXl0Zbk8_rvjyLwAR-Xh9pQ >> >> With an algebraic approach to differential geometry, the >> possibility of working over finite fields emerges. This is another key advantage to following Newton, Euler and Lagrange when it comes to calculus! >> >> In this lecture we introduce the basics of finite (prime) fields, where we work mod p for some fixed prime p, and show that our study of tangent conics to a cubic polynomial extends naturally, and leads to interesting combinatorial structures. There are many possible directions for investigation by interested amateurs who have understood this lecture. >> >> After the basics of arithmetic over the field F_p, including a discussion of primitive roots and Fermat's theorem, we discuss polynomial arithmetic and illustrate tangent conics to a particular cubic over F_11. In particular Ghys' lovely observation about the disjointness of such tangent conics (for a cubic) can be illustrated completely here, and some additional patterns visibly emerge from the vertices of the various tangent conics. >> >> One big difference here is that the sub-derivatives and the derivatives are NOT equivalent in general, and we must replace the usual Taylor expansions with one involving sub-derivatives. Some remarks about the useful distinction between polynomials and polynomial functions in this setting are made. >> >> This lecture shows that the calculus is actually a much wider operational tool than is usually appreciated----finite calculus not only makes sense but is a rich source of both combinatorial and algebraic patterns---and questions for further investigations. > > Andy.Latto@pobox.com
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participants (5)
-
Christian Lawson-Perfect -
Fred Lunnon -
Henry Baker -
James Propp -
Neil Sloane