[math-fun] Gimbal lock??
Perhaps it's time to learn quaternions??
From comp.risks:
Date: Fri, 10 Jan 2020 20:24:07 +0000 From: "Clive D.W. Feather" <clive@davros.org> Subject: Boeing 737s can't land facing west (FAA) "The FAA received reports earlier this year of three incidents of display electronic unit (DEU) software errors on Model 737 NG airplanes flying into runway PABR in Barrow, Alaska. All six display units (DUs) blanked with a selected instrument approach to a runway with a 270-degree true heading, and all six DUs stayed blank until a different runway was selected. [...] The investigation revealed that the problem occurs when this combination of software is installed and a susceptible runway with a 270-degree true heading is selected for instrument approach. Not all runways with a 270-degree true heading are susceptible; only seven runways worldwide, as identified in this AD, have latitude and longitude values that cause the blanking behavior." (Note that this is all 6 displays on each plane, not 2 displays on each of three planes.) The runways in question are: Runway 26, Pine Bluffs, Wyoming, USA (82V) Runway 28, Wayne County, Ohio, USA (KBJJ) Runway 28, Chippewa County, Michigan, USA (KCIU) Runway 26, Cavern City, New Mexico, USA (KCNM) Runway 25, Barrow, Alaska, USA (PABR) Runway 28, La Mina, La Guajira, Colombia (SKLM) Runway 29, Cheddi Jagan, Georgetown, Guyana (SYCJ) (The numbers are magnetic bearings, whereas the problem is apparently related to true bearing.) Original FAA notice: <http://rgl.faa.gov/Regulatory_and_Guidance_Library/rgad.nsf/0/3948342a978cc27b862584dd005c1a60/$FILE/2019-25-17.pdf>
Good luck with convincing the engineers involved! Mind you, it is only getting on for 180 years ago that Hamilton came up with quaternions. So early days ... er, centuries? But perhaps it might be more constructive to attempt to understand why there is such instinctive resistance among engineers to the whole notion of quaternions, never mind more general geometric (Clifford) algebra. There's a frustrating pedagogical phenomenon involved in such investigations. While I can well recall my own sense of bewilderment on first encountering Hestenes' early book on these matters, I cannot muster the slightest insight into the cause of those conceptual difficulties. As a result, I now am stranded as far away from offering assistance to the uninitiated as I earlier was from receiving any: I simply cannot understand _why_ they can't understand. Anyway, here's a couple of possible clues: perhaps others can come up with more suggestions. (A) It's noteworthy that the first thing Heaviside did was to dissect Hamiton's elegant unity into "scalar" & "vector" parts, which went on to gain pretty much universal acceptance. There seems to be a mental hurdle in human minds obstructing the union of disparate familiar categories under a common umbrella: in this case, familiarity with angles & Cartesian coordinates actively obstructs conceptualisation of quaternions. (B) It is rarely made explicit that (like vectors) quaternions come in two flavours: polar & axial, depending on application. It seems that Hamilton himself was confused over this, which contributed to early controversy about their validity. There's a informative but slightly muddled paper on this topic somewhere on the internet which proceeds from the quaint premiss that they must exclusively be one or the other, despite the arguments put forward clearly illustrating a dichotomy. WFL On 1/28/20, Henry Baker <hbaker1@pipeline.com> wrote:
Perhaps it's time to learn quaternions??
From comp.risks:
Date: Fri, 10 Jan 2020 20:24:07 +0000 From: "Clive D.W. Feather" <clive@davros.org> Subject: Boeing 737s can't land facing west (FAA)
"The FAA received reports earlier this year of three incidents of display electronic unit (DEU) software errors on Model 737 NG airplanes flying into runway PABR in Barrow, Alaska. All six display units (DUs) blanked with a selected instrument approach to a runway with a 270-degree true heading, and all six DUs stayed blank until a different runway was selected. [...] The investigation revealed that the problem occurs when this combination of software is installed and a susceptible runway with a 270-degree true heading is selected for instrument approach. Not all runways with a 270-degree true heading are susceptible; only seven runways worldwide, as identified in this AD, have latitude and longitude values that cause the blanking behavior."
(Note that this is all 6 displays on each plane, not 2 displays on each of three planes.)
The runways in question are:
Runway 26, Pine Bluffs, Wyoming, USA (82V) Runway 28, Wayne County, Ohio, USA (KBJJ) Runway 28, Chippewa County, Michigan, USA (KCIU) Runway 26, Cavern City, New Mexico, USA (KCNM) Runway 25, Barrow, Alaska, USA (PABR) Runway 28, La Mina, La Guajira, Colombia (SKLM) Runway 29, Cheddi Jagan, Georgetown, Guyana (SYCJ)
(The numbers are magnetic bearings, whereas the problem is apparently related to true bearing.)
Original FAA notice: <http://rgl.faa.gov/Regulatory_and_Guidance_Library/rgad.nsf/0/3948342a978cc27b862584dd005c1a60/$FILE/2019-25-17.pdf>
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A huge disappointment from my MIT math and physics education was the lack of explanation early on of the difference between vectors and co-vectors. It wasn’t until I started reading about differential geometry that I realized the pervasiveness of this education-driven ignorance. All of a sudden lots of things that never made sense started clicking. How and why do we allow this fuzziness to continue?
On Jan 28, 2020, at 11:16 PM, Fred Lunnon <fred.lunnon@gmail.com> wrote:
Good luck with convincing the engineers involved! Mind you, it is only getting on for 180 years ago that Hamilton came up with quaternions. So early days ... er, centuries?
But perhaps it might be more constructive to attempt to understand why there is such instinctive resistance among engineers to the whole notion of quaternions, never mind more general geometric (Clifford) algebra.
There's a frustrating pedagogical phenomenon involved in such investigations. While I can well recall my own sense of bewilderment on first encountering Hestenes' early book on these matters, I cannot muster the slightest insight into the cause of those conceptual difficulties. As a result, I now am stranded as far away from offering assistance to the uninitiated as I earlier was from receiving any: I simply cannot understand _why_ they can't understand.
Anyway, here's a couple of possible clues: perhaps others can come up with more suggestions.
(A) It's noteworthy that the first thing Heaviside did was to dissect Hamiton's elegant unity into "scalar" & "vector" parts, which went on to gain pretty much universal acceptance. There seems to be a mental hurdle in human minds obstructing the union of disparate familiar categories under a common umbrella: in this case, familiarity with angles & Cartesian coordinates actively obstructs conceptualisation of quaternions.
(B) It is rarely made explicit that (like vectors) quaternions come in two flavours: polar & axial, depending on application. It seems that Hamilton himself was confused over this, which contributed to early controversy about their validity. There's a informative but slightly muddled paper on this topic somewhere on the internet which proceeds from the quaint premiss that they must exclusively be one or the other, despite the arguments put forward clearly illustrating a dichotomy.
WFL
On 1/28/20, Henry Baker <hbaker1@pipeline.com> wrote:
Perhaps it's time to learn quaternions??
From comp.risks:
Date: Fri, 10 Jan 2020 20:24:07 +0000 From: "Clive D.W. Feather" <clive@davros.org> Subject: Boeing 737s can't land facing west (FAA)
"The FAA received reports earlier this year of three incidents of display electronic unit (DEU) software errors on Model 737 NG airplanes flying into runway PABR in Barrow, Alaska. All six display units (DUs) blanked with a selected instrument approach to a runway with a 270-degree true heading, and all six DUs stayed blank until a different runway was selected. [...] The investigation revealed that the problem occurs when this combination of software is installed and a susceptible runway with a 270-degree true heading is selected for instrument approach. Not all runways with a 270-degree true heading are susceptible; only seven runways worldwide, as identified in this AD, have latitude and longitude values that cause the blanking behavior."
(Note that this is all 6 displays on each plane, not 2 displays on each of three planes.)
The runways in question are:
Runway 26, Pine Bluffs, Wyoming, USA (82V) Runway 28, Wayne County, Ohio, USA (KBJJ) Runway 28, Chippewa County, Michigan, USA (KCIU) Runway 26, Cavern City, New Mexico, USA (KCNM) Runway 25, Barrow, Alaska, USA (PABR) Runway 28, La Mina, La Guajira, Colombia (SKLM) Runway 29, Cheddi Jagan, Georgetown, Guyana (SYCJ)
(The numbers are magnetic bearings, whereas the problem is apparently related to true bearing.)
Original FAA notice: <http://rgl.faa.gov/Regulatory_and_Guidance_Library/rgad.nsf/0/3948342a978cc27b862584dd005c1a60/$FILE/2019-25-17.pdf>
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The flavor of the distinction is very similar to unit analysis from physics, which tags some scalars with "types" which behave coherently under arithmetic. In principle, you are never allowed to add two scalars of differing types. Presumably, whenever you see a dot product, there is a vector on the right and a covector on the left, and it is a type error to add a vector and a covector. Do these types really behave consistently, or are there exceptions to the strict classification? On Wed, Jan 29, 2020 at 10:47 AM Tom Knight <tk@mit.edu> wrote:
A huge disappointment from my MIT math and physics education was the lack of explanation early on of the difference between vectors and co-vectors. It wasn’t until I started reading about differential geometry that I realized the pervasiveness of this education-driven ignorance. All of a sudden lots of things that never made sense started clicking. How and why do we allow this fuzziness to continue?
On Jan 28, 2020, at 11:16 PM, Fred Lunnon <fred.lunnon@gmail.com> wrote:
Good luck with convincing the engineers involved! Mind you, it is only getting on for 180 years ago that Hamilton came up with quaternions. So early days ... er, centuries?
But perhaps it might be more constructive to attempt to understand why there is such instinctive resistance among engineers to the whole notion of quaternions, never mind more general geometric (Clifford) algebra.
There's a frustrating pedagogical phenomenon involved in such investigations. While I can well recall my own sense of bewilderment on first encountering Hestenes' early book on these matters, I cannot muster the slightest insight into the cause of those conceptual difficulties. As a result, I now am stranded as far away from offering assistance to the uninitiated as I earlier was from receiving any: I simply cannot understand _why_ they can't understand.
Anyway, here's a couple of possible clues: perhaps others can come up with more suggestions.
(A) It's noteworthy that the first thing Heaviside did was to dissect Hamiton's elegant unity into "scalar" & "vector" parts, which went on to gain pretty much universal acceptance. There seems to be a mental hurdle in human minds obstructing the union of disparate familiar categories under a common umbrella: in this case, familiarity with angles & Cartesian coordinates actively obstructs conceptualisation of quaternions.
(B) It is rarely made explicit that (like vectors) quaternions come in two flavours: polar & axial, depending on application. It seems that Hamilton himself was confused over this, which contributed to early controversy about their validity. There's a informative but slightly muddled paper on this topic somewhere on the internet which proceeds from the quaint premiss that they must exclusively be one or the other, despite the arguments put forward clearly illustrating a dichotomy.
WFL
On 1/28/20, Henry Baker <hbaker1@pipeline.com> wrote:
Perhaps it's time to learn quaternions??
From comp.risks:
Date: Fri, 10 Jan 2020 20:24:07 +0000 From: "Clive D.W. Feather" <clive@davros.org> Subject: Boeing 737s can't land facing west (FAA)
"The FAA received reports earlier this year of three incidents of display electronic unit (DEU) software errors on Model 737 NG airplanes flying into runway PABR in Barrow, Alaska. All six display units (DUs) blanked with a selected instrument approach to a runway with a 270-degree true heading, and all six DUs stayed blank until a different runway was selected. [...] The investigation revealed that the problem occurs when this combination of software is installed and a susceptible runway with a 270-degree true heading is selected for instrument approach. Not all runways with a 270-degree true heading are susceptible; only seven runways worldwide, as identified in this AD, have latitude and longitude values that cause the blanking behavior."
(Note that this is all 6 displays on each plane, not 2 displays on each of three planes.)
The runways in question are:
Runway 26, Pine Bluffs, Wyoming, USA (82V) Runway 28, Wayne County, Ohio, USA (KBJJ) Runway 28, Chippewa County, Michigan, USA (KCIU) Runway 26, Cavern City, New Mexico, USA (KCNM) Runway 25, Barrow, Alaska, USA (PABR) Runway 28, La Mina, La Guajira, Colombia (SKLM) Runway 29, Cheddi Jagan, Georgetown, Guyana (SYCJ)
(The numbers are magnetic bearings, whereas the problem is apparently related to true bearing.)
Original FAA notice: < http://rgl.faa.gov/Regulatory_and_Guidance_Library/rgad.nsf/0/3948342a978cc2...
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For types I say "dimensions" as in "Dimensional Analysis". A major teaching frustration is that most students can barely handle units, much less dimensions. Then they ask: "What is Electric Flux?", and can't understand the answer that it is a measurable quantity of dimension [F][L]^2 / [Q]. I would like to write out some of my thoughts about how dimensional analysis is a lot like projective geometry, but for now, do not have the time. ( And did Max Planck think about this?) As for co-vectors, vectors, linear functionals etc., If the elements are complex numbers--as they are in quantum-- there will be type errors on mismatched addition, because norming requires complex conjugation. --Brad On Wed, Jan 29, 2020 at 10:11 AM Allan Wechsler <acwacw@gmail.com> wrote:
The flavor of the distinction is very similar to unit analysis from physics, which tags some scalars with "types" which behave coherently under arithmetic. In principle, you are never allowed to add two scalars of differing types.
Presumably, whenever you see a dot product, there is a vector on the right and a covector on the left, and it is a type error to add a vector and a covector. Do these types really behave consistently, or are there exceptions to the strict classification?
On Wed, Jan 29, 2020 at 10:47 AM Tom Knight <tk@mit.edu> wrote:
A huge disappointment from my MIT math and physics education was the lack of explanation early on of the difference between vectors and co-vectors. It wasn’t until I started reading about differential geometry that I realized the pervasiveness of this education-driven ignorance. All of a sudden lots of things that never made sense started clicking. How and why do we allow this fuzziness to continue?
On Jan 28, 2020, at 11:16 PM, Fred Lunnon <fred.lunnon@gmail.com> wrote:
Good luck with convincing the engineers involved! Mind you, it is only getting on for 180 years ago that Hamilton came up with quaternions. So early days ... er, centuries?
But perhaps it might be more constructive to attempt to understand why there is such instinctive resistance among engineers to the whole notion of quaternions, never mind more general geometric (Clifford) algebra.
There's a frustrating pedagogical phenomenon involved in such investigations. While I can well recall my own sense of bewilderment on first encountering Hestenes' early book on these matters, I cannot muster the slightest insight into the cause of those conceptual difficulties. As a result, I now am stranded as far away from offering assistance to the uninitiated as I earlier was from receiving any: I simply cannot understand _why_ they can't understand.
Anyway, here's a couple of possible clues: perhaps others can come up with more suggestions.
(A) It's noteworthy that the first thing Heaviside did was to dissect Hamiton's elegant unity into "scalar" & "vector" parts, which went on to gain pretty much universal acceptance. There seems to be a mental hurdle in human minds obstructing the union of disparate familiar categories under a common umbrella: in this case, familiarity with angles & Cartesian coordinates actively obstructs conceptualisation of quaternions.
(B) It is rarely made explicit that (like vectors) quaternions come in two flavours: polar & axial, depending on application. It seems that Hamilton himself was confused over this, which contributed to early controversy about their validity. There's a informative but slightly muddled paper on this topic somewhere on the internet which proceeds from the quaint premiss that they must exclusively be one or the other, despite the arguments put forward clearly illustrating a dichotomy.
WFL
On 1/28/20, Henry Baker <hbaker1@pipeline.com> wrote:
Perhaps it's time to learn quaternions??
From comp.risks:
Date: Fri, 10 Jan 2020 20:24:07 +0000 From: "Clive D.W. Feather" <clive@davros.org> Subject: Boeing 737s can't land facing west (FAA)
"The FAA received reports earlier this year of three incidents of display electronic unit (DEU) software errors on Model 737 NG airplanes flying into runway PABR in Barrow, Alaska. All six display units (DUs) blanked with a selected instrument approach to a runway with a 270-degree true heading, and all six DUs stayed blank until a different runway was selected. [...] The investigation revealed that the problem occurs when this combination of software is installed and a susceptible runway with a 270-degree true heading is selected for instrument approach. Not all runways with a 270-degree true heading are susceptible; only seven runways worldwide, as identified in this AD, have latitude and longitude values that cause the blanking behavior."
(Note that this is all 6 displays on each plane, not 2 displays on each of three planes.)
The runways in question are:
Runway 26, Pine Bluffs, Wyoming, USA (82V) Runway 28, Wayne County, Ohio, USA (KBJJ) Runway 28, Chippewa County, Michigan, USA (KCIU) Runway 26, Cavern City, New Mexico, USA (KCNM) Runway 25, Barrow, Alaska, USA (PABR) Runway 28, La Mina, La Guajira, Colombia (SKLM) Runway 29, Cheddi Jagan, Georgetown, Guyana (SYCJ)
(The numbers are magnetic bearings, whereas the problem is apparently related to true bearing.)
Original FAA notice: < http://rgl.faa.gov/Regulatory_and_Guidance_Library/rgad.nsf/0/3948342a978cc2...
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There's a very nice book by George Hart, "Multidimensional Analysis" which brings some order and rigor to the often sloppy use of dimensioned quantities by engineers. He develops a linear algebra of dimensioned matrices. Brent On 1/29/2020 8:55 AM, Brad Klee wrote:
For types I say "dimensions" as in "Dimensional Analysis". A major teaching frustration is that most students can barely handle units, much less dimensions. Then they ask: "What is Electric Flux?", and can't understand the answer that it is a measurable quantity of dimension [F][L]^2 / [Q].
I would like to write out some of my thoughts about how dimensional analysis is a lot like projective geometry, but for now, do not have the time. ( And did Max Planck think about this?)
As for co-vectors, vectors, linear functionals etc., If the elements are complex numbers--as they are in quantum-- there will be type errors on mismatched addition, because norming requires complex conjugation.
--Brad
On Wed, Jan 29, 2020 at 10:11 AM Allan Wechsler <acwacw@gmail.com> wrote:
The flavor of the distinction is very similar to unit analysis from physics, which tags some scalars with "types" which behave coherently under arithmetic. In principle, you are never allowed to add two scalars of differing types.
Presumably, whenever you see a dot product, there is a vector on the right and a covector on the left, and it is a type error to add a vector and a covector. Do these types really behave consistently, or are there exceptions to the strict classification?
On Wed, Jan 29, 2020 at 10:47 AM Tom Knight <tk@mit.edu> wrote:
A huge disappointment from my MIT math and physics education was the lack of explanation early on of the difference between vectors and co-vectors. It wasn’t until I started reading about differential geometry that I realized the pervasiveness of this education-driven ignorance. All of a sudden lots of things that never made sense started clicking. How and why do we allow this fuzziness to continue?
On Jan 28, 2020, at 11:16 PM, Fred Lunnon <fred.lunnon@gmail.com> wrote:
Good luck with convincing the engineers involved! Mind you, it is only getting on for 180 years ago that Hamilton came up with quaternions. So early days ... er, centuries?
But perhaps it might be more constructive to attempt to understand why there is such instinctive resistance among engineers to the whole notion of quaternions, never mind more general geometric (Clifford) algebra.
There's a frustrating pedagogical phenomenon involved in such investigations. While I can well recall my own sense of bewilderment on first encountering Hestenes' early book on these matters, I cannot muster the slightest insight into the cause of those conceptual difficulties. As a result, I now am stranded as far away from offering assistance to the uninitiated as I earlier was from receiving any: I simply cannot understand _why_ they can't understand.
Anyway, here's a couple of possible clues: perhaps others can come up with more suggestions.
(A) It's noteworthy that the first thing Heaviside did was to dissect Hamiton's elegant unity into "scalar" & "vector" parts, which went on to gain pretty much universal acceptance. There seems to be a mental hurdle in human minds obstructing the union of disparate familiar categories under a common umbrella: in this case, familiarity with angles & Cartesian coordinates actively obstructs conceptualisation of quaternions.
(B) It is rarely made explicit that (like vectors) quaternions come in two flavours: polar & axial, depending on application. It seems that Hamilton himself was confused over this, which contributed to early controversy about their validity. There's a informative but slightly muddled paper on this topic somewhere on the internet which proceeds from the quaint premiss that they must exclusively be one or the other, despite the arguments put forward clearly illustrating a dichotomy.
WFL
On 1/28/20, Henry Baker <hbaker1@pipeline.com> wrote:
Perhaps it's time to learn quaternions??
From comp.risks:
Date: Fri, 10 Jan 2020 20:24:07 +0000 From: "Clive D.W. Feather" <clive@davros.org> Subject: Boeing 737s can't land facing west (FAA)
"The FAA received reports earlier this year of three incidents of display electronic unit (DEU) software errors on Model 737 NG airplanes flying into runway PABR in Barrow, Alaska. All six display units (DUs) blanked with a selected instrument approach to a runway with a 270-degree true heading, and all six DUs stayed blank until a different runway was selected. [...] The investigation revealed that the problem occurs when this combination of software is installed and a susceptible runway with a 270-degree true heading is selected for instrument approach. Not all runways with a 270-degree true heading are susceptible; only seven runways worldwide, as identified in this AD, have latitude and longitude values that cause the blanking behavior."
(Note that this is all 6 displays on each plane, not 2 displays on each of three planes.)
The runways in question are:
Runway 26, Pine Bluffs, Wyoming, USA (82V) Runway 28, Wayne County, Ohio, USA (KBJJ) Runway 28, Chippewa County, Michigan, USA (KCIU) Runway 26, Cavern City, New Mexico, USA (KCNM) Runway 25, Barrow, Alaska, USA (PABR) Runway 28, La Mina, La Guajira, Colombia (SKLM) Runway 29, Cheddi Jagan, Georgetown, Guyana (SYCJ)
(The numbers are magnetic bearings, whereas the problem is apparently related to true bearing.)
Original FAA notice: < http://rgl.faa.gov/Regulatory_and_Guidance_Library/rgad.nsf/0/3948342a978cc2...
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Trig provides another arena in which there are different types of scalars. Angles and ratios-of-lengths are scalars, but in applications of trig it makes no sense to add an angle to the sine of another angle; angles and ratios are different types of numbers. Has anyone worked out a framework for this perspective? (I don’t think George treats trig in his book, though it’s possible that he did and I’m forgetting it.) Such a framework would rewrite the standard power series formula for sin x as sin x = r x / 1! - r^3 x^3 / 3! + ... where r is a type-conversion coefficient. What would the point of such a framework be (besides an idiosyncratic form of ideological purity)? Well, think of all those formulas in which some integral is shown to equal some rational coefficient times pi to the power of some rational exponent. Wouldn't it be nice to have a “dimensional” way to predict the exponent? Jim Propp On Wed, Jan 29, 2020 at 5:32 PM Brent Meeker via math-fun < math-fun@mailman.xmission.com> wrote:
There's a very nice book by George Hart, "Multidimensional Analysis" which brings some order and rigor to the often sloppy use of dimensioned quantities by engineers. He develops a linear algebra of dimensioned matrices.
Brent
On 1/29/2020 8:55 AM, Brad Klee wrote:
For types I say "dimensions" as in "Dimensional Analysis". A major teaching frustration is that most students can barely handle units, much less dimensions. Then they ask: "What is Electric Flux?", and can't understand the answer that it is a measurable quantity of dimension [F][L]^2 / [Q].
I would like to write out some of my thoughts about how dimensional analysis is a lot like projective geometry, but for now, do not have the time. ( And did Max Planck think about this?)
As for co-vectors, vectors, linear functionals etc., If the elements are complex numbers--as they are in quantum-- there will be type errors on mismatched addition, because norming requires complex conjugation.
--Brad
On Wed, Jan 29, 2020 at 10:11 AM Allan Wechsler <acwacw@gmail.com> wrote:
The flavor of the distinction is very similar to unit analysis from physics, which tags some scalars with "types" which behave coherently under arithmetic. In principle, you are never allowed to add two scalars of differing types.
Presumably, whenever you see a dot product, there is a vector on the right and a covector on the left, and it is a type error to add a vector and a covector. Do these types really behave consistently, or are there exceptions to the strict classification?
On Wed, Jan 29, 2020 at 10:47 AM Tom Knight <tk@mit.edu> wrote:
A huge disappointment from my MIT math and physics education was the lack of explanation early on of the difference between vectors and co-vectors. It wasn’t until I started reading about differential geometry that I realized the pervasiveness of this education-driven ignorance. All of a sudden lots of things that never made sense started clicking. How and why do we allow this fuzziness to continue?
On Jan 28, 2020, at 11:16 PM, Fred Lunnon <fred.lunnon@gmail.com> wrote:
Good luck with convincing the engineers involved! Mind you, it is only getting on for 180 years ago that Hamilton came up with quaternions. So early days ... er, centuries?
But perhaps it might be more constructive to attempt to understand why there is such instinctive resistance among engineers to the whole notion of quaternions, never mind more general geometric (Clifford) algebra.
There's a frustrating pedagogical phenomenon involved in such investigations. While I can well recall my own sense of bewilderment on first encountering Hestenes' early book on these matters, I cannot muster the slightest insight into the cause of those conceptual difficulties. As a result, I now am stranded as far away from offering assistance to the uninitiated as I earlier was from receiving any: I simply cannot understand _why_ they can't understand.
Anyway, here's a couple of possible clues: perhaps others can come up with more suggestions.
(A) It's noteworthy that the first thing Heaviside did was to dissect Hamiton's elegant unity into "scalar" & "vector" parts, which went on to gain pretty much universal acceptance. There seems to be a mental hurdle in human minds obstructing the union of disparate familiar categories under a common umbrella: in this case, familiarity with angles & Cartesian coordinates actively obstructs conceptualisation of quaternions.
(B) It is rarely made explicit that (like vectors) quaternions come in two flavours: polar & axial, depending on application. It seems that Hamilton himself was confused over this, which contributed to early controversy about their validity. There's a informative but slightly muddled paper on this topic somewhere on the internet which proceeds from the quaint premiss that they must exclusively be one or the other, despite the arguments put forward clearly illustrating a dichotomy.
WFL
On 1/28/20, Henry Baker <hbaker1@pipeline.com> wrote:
Perhaps it's time to learn quaternions??
From comp.risks:
Date: Fri, 10 Jan 2020 20:24:07 +0000 From: "Clive D.W. Feather" <clive@davros.org> Subject: Boeing 737s can't land facing west (FAA)
"The FAA received reports earlier this year of three incidents of display electronic unit (DEU) software errors on Model 737 NG airplanes flying into runway PABR in Barrow, Alaska. All six display units (DUs) blanked with a selected instrument approach to a runway with a 270-degree true heading, and all six DUs stayed blank until a different runway was selected. [...] The investigation revealed that the problem occurs when this combination of software is installed and a susceptible runway with a 270-degree true heading is selected for instrument approach. Not all runways with a 270-degree true heading are susceptible; only seven runways worldwide, as identified in this AD, have latitude and longitude values that cause the blanking behavior."
(Note that this is all 6 displays on each plane, not 2 displays on each of three planes.)
The runways in question are:
Runway 26, Pine Bluffs, Wyoming, USA (82V) Runway 28, Wayne County, Ohio, USA (KBJJ) Runway 28, Chippewa County, Michigan, USA (KCIU) Runway 26, Cavern City, New Mexico, USA (KCNM) Runway 25, Barrow, Alaska, USA (PABR) Runway 28, La Mina, La Guajira, Colombia (SKLM) Runway 29, Cheddi Jagan, Georgetown, Guyana (SYCJ)
(The numbers are magnetic bearings, whereas the problem is apparently related to true bearing.)
Original FAA notice: <
http://rgl.faa.gov/Regulatory_and_Guidance_Library/rgad.nsf/0/3948342a978cc2...
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In Kepler's Equation it does make sense to set Area = x + e*sin(x). This area can be constructed by compass and straightedge as an n-gon, with n -> Infinity. Kepler must have thought of this equation in terms of trig, because modern trig. function theory essentially starts with Euler. As for your question about pi-dimensions, it's difficult to interpret what you are asking for without explicit examples. For some integrals, it is easy to predict, the output pi exponent, see for example [1]. A major preventative difficulty to the dimensional analysis of integrals is well stated in K-Z monograph on periods, Conjecture 1: "If a period has two integral representations, then one can pass from one formula to another using only [additivity, change of variables, and anti- differentiation] in which all functions and domains of integration are algebraic with coefficients in Q." The situation is only worse when considering larger classes of integrable functions. Before a comprehensive dimensional system could be developed for pi integrals, I think that the transformation theory needs more work. This Conjecture 1 (and possibly more) would probably need to be settled. --Brad [1] https://mathoverflow.net/questions/328657/why-these-surprising-proportionali... On Thu, Jan 30, 2020 at 5:14 AM James Propp <jamespropp@gmail.com> wrote:
Trig provides another arena in which there are different types of scalars. Angles and ratios-of-lengths are scalars, but in applications of trig it makes no sense to add an angle to the sine of another angle; angles and ratios are different types of numbers.
Has anyone worked out a framework for this perspective? (I don’t think George treats trig in his book, though it’s possible that he did and I’m forgetting it.)
Such a framework would rewrite the standard power series formula for sin x as sin x = r x / 1! - r^3 x^3 / 3! + ... where r is a type-conversion coefficient.
What would the point of such a framework be (besides an idiosyncratic form of ideological purity)? Well, think of all those formulas in which some integral is shown to equal some rational coefficient times pi to the power of some rational exponent. Wouldn't it be nice to have a “dimensional” way to predict the exponent?
Jim Propp
Jim, The case of angles and trig functions are good for bringing out the fuzziness in most people's basic conception of physical dimension. Angles seem dimensionless if you think of them as a quotient of a length with a length (as in the definition of a radian), but they seem dimensioned if you think of how it can be useful to change the fundamental unit (e.g., degrees, radians, grads, cycles...) or if you consider that you can do a physical experiment to add two angles and observe their sum. Because the power series for Cosine only uses even powers, it has been proposed that to avoid the kind of unmotivated conversion factor (r) you put in your Sine series, the dimensions of angle could have the property of not being dimensionless, yet being a square root of dimensionless, i.e., like a square root of identity other than identity. (You would still need one factor in the Sine series, because it is the odd powers.) Can the algebra of a physical dimension be a finite cyclic group in this way? Might square roots not be dimensionally unique? Is it meaningful to ask if someday an experiment might show that mass^10=dimensionless? Is (1 meter)^Sqrt[2] defined? Is (1 meter)^Sqrt[-1] defined? I find that even people who work with dimensions all the time can have surprisingly different feelings about such questions and what the structure of the algebra of dimensions might be. As with angle and dimensionlessness, torque and energy appear to have the have the same exponents in their dimensional terms (length^2 * mass^1 * time^-2 * charge^0) yet different physical roles in which they are never added, suggesting standard algebras of dimensions do not quite capture everything of interest even for defining something as basic as when addition of two quantities is defined. These sorts of issues pop up in my book, but are not the central focus. As far as I know, no one has a fully satisfactory framework even for scalars. There is another level of richness when considering the dimensions of vectors, matrices, etc. But to the original point: including dimensions within the linear algebra brings out very clearly the difference between a vector and a co-vector. George http://georgehart.com On 1/30/2020 6:14 AM, James Propp wrote:
Trig provides another arena in which there are different types of scalars. Angles and ratios-of-lengths are scalars, but in applications of trig it makes no sense to add an angle to the sine of another angle; angles and ratios are different types of numbers.
Has anyone worked out a framework for this perspective? (I don’t think George treats trig in his book, though it’s possible that he did and I’m forgetting it.)
Such a framework would rewrite the standard power series formula for sin x as sin x = r x / 1! - r^3 x^3 / 3! + ... where r is a type-conversion coefficient.
What would the point of such a framework be (besides an idiosyncratic form of ideological purity)? Well, think of all those formulas in which some integral is shown to equal some rational coefficient times pi to the power of some rational exponent. Wouldn't it be nice to have a “dimensional” way to predict the exponent?
Jim Propp
On Wed, Jan 29, 2020 at 5:32 PM Brent Meeker via math-fun < math-fun@mailman.xmission.com> wrote:
There's a very nice book by George Hart, "Multidimensional Analysis" which brings some order and rigor to the often sloppy use of dimensioned quantities by engineers. He develops a linear algebra of dimensioned matrices.
Brent
On 1/29/2020 8:55 AM, Brad Klee wrote:
For types I say "dimensions" as in "Dimensional Analysis". A major teaching frustration is that most students can barely handle units, much less dimensions. Then they ask: "What is Electric Flux?", and can't understand the answer that it is a measurable quantity of dimension [F][L]^2 / [Q].
I would like to write out some of my thoughts about how dimensional analysis is a lot like projective geometry, but for now, do not have the time. ( And did Max Planck think about this?)
As for co-vectors, vectors, linear functionals etc., If the elements are complex numbers--as they are in quantum-- there will be type errors on mismatched addition, because norming requires complex conjugation.
--Brad
On Wed, Jan 29, 2020 at 10:11 AM Allan Wechsler <acwacw@gmail.com> wrote:
The flavor of the distinction is very similar to unit analysis from physics, which tags some scalars with "types" which behave coherently under arithmetic. In principle, you are never allowed to add two scalars of differing types.
Presumably, whenever you see a dot product, there is a vector on the right and a covector on the left, and it is a type error to add a vector and a covector. Do these types really behave consistently, or are there exceptions to the strict classification?
...
In my experience, the most common response is a reminder that: "If you want to accomplish practical work in science and engineering, you shouldn't waste time on corner cases, such as fractal curves etc." This could be regional bias. The problem you mention about torque and energy, I don't see it as a dimensional problem, but rather as arising from macroscopic / microscopic divide. Torque should actually be lumped in with force. Similarly it is possible to show, using Lagrange multipliers, that angular and linear momentum can conserve together, see for example [1]. I would think that these sort of extra [L] factors usually arise from working with extensive bodies. As for your assertion about co-vectors, I'm skeptical and have not had time to read your book. One counter argument is that dimensions should be swept into the metric itself. In Dirac notation, if (v_1|v_2) = (v_1|g_0|v_2), then change of basis can be accomplished by g_0 -> g_1 = k_1*k_2*g_0, then (v_1|g_1|v_2) = k_1*k_2*(v_1|v_2), with k1, k2 dimensional factors. However, a stronger counterargument is to entirely forget about linear algebra, and to propose another model. Here is a brief description of the projective system I mentioned earlier: The real projective line has coordinates [X:Y] with absolute magnitude M=X/Y. Let us associate Y=1 with a fixed scale of measurement, for example Cesium's hyperfine frequency split, 9192631770 Hz, and let us associate X with, say, the frequency of an electric dipole oscillating in a constant electric field. One acceptable definition of measurement is that for any possible measurement of X, the absolute value of M must remain constant. The projective coordinate system allows covariant change of Y, i.e. change of units. In practice, the unit system [X:1] is very rarely used, with Hz a more common variant, though less precise. To change units we have to solve the projective equation [1:1] = [9192631770:Y'], thus Y' = 1/9192631770. Now a measurement of X' in Hz has a representation [X':1/9192631770], whose absolute value is equal to another measurement in the [x:1] system. It is clear that any number Y, Y', Y'' etc. is a dimensional unit, for example: 1 = 1 Caesium cycle / 9192631770 Hz, 1 = 1 KHz / 1000 Hz, etc. In teaching I would almost never mention the analogy to projective geometry, but always emphasize the point that changing units is equivalent to multiplication by unity, a dimensionless factor. I plan to write all this up more clearly in the future. That could be a chance for me to look through your book or to think about corner cases, but as I said, I'm initially skeptical. Cheers, --Brad [1] http://physics.princeton.edu/~mcdonald/examples/2cylinders.pdf Historical note: This problem was not "originally suggested" by me, it was "originally solved" by me. Kirk's first solution, which accompanied a journal rejection, was incorrect. I corrected him, and he published his subsequent work online. I have the emails. * * * They owe me a cracker at Princeton! * * * On Thu, Jan 30, 2020 at 9:57 PM George Hart <george@georgehart.com> wrote:
I find that even people who work with dimensions all the time can have surprisingly different feelings about such questions and what the structure of the algebra of dimensions might be.
As with angle and dimensionlessness, torque and energy appear to have the have the same exponents in their dimensional terms (length^2 * mass^1 * time^-2 * charge^0) yet different physical roles in which they are never added, suggesting standard algebras of dimensions do not quite capture everything of interest even for defining something as basic as when addition of two quantities is defined.
These sorts of issues pop up in my book, but are not the central focus. As far as I know, no one has a fully satisfactory framework even for scalars. There is another level of richness when considering the dimensions of vectors, matrices, etc. But to the original point: including dimensions within the linear algebra brings out very clearly the difference between a vector and a co-vector.
George http://georgehart.com
Arg. Y'=9192631770, with Hz system [X':9192631770] relative to [X:1] on the Caesium standard. Sorry for the multi-multi-post... On Fri, Jan 31, 2020 at 9:53 AM Brad Klee <bradklee@gmail.com> wrote:
In my experience, the most common response is a reminder that: "If you want to accomplish practical work in science and engineering, you shouldn't waste time on corner cases, such as fractal curves etc." This could be regional bias.
The problem you mention about torque and energy, I don't see it as a dimensional problem, but rather as arising from macroscopic / microscopic divide. Torque should actually be lumped in with force. Similarly it is possible to show, using Lagrange multipliers, that angular and linear momentum can conserve together, see for example [1]. I would think that these sort of extra [L] factors usually arise from working with extensive bodies.
As for your assertion about co-vectors, I'm skeptical and have not had time to read your book. One counter argument is that dimensions should be swept into the metric itself. In Dirac notation, if (v_1|v_2) = (v_1|g_0|v_2), then change of basis can be accomplished by g_0 -> g_1 = k_1*k_2*g_0, then (v_1|g_1|v_2) = k_1*k_2*(v_1|v_2), with k1, k2 dimensional factors.
However, a stronger counterargument is to entirely forget about linear algebra, and to propose another model. Here is a brief description of the projective system I mentioned earlier:
The real projective line has coordinates [X:Y] with absolute magnitude M=X/Y. Let us associate Y=1 with a fixed scale of measurement, for example Cesium's hyperfine frequency split, 9192631770 Hz, and let us associate X with, say, the frequency of an electric dipole oscillating in a constant electric field. One acceptable definition of measurement is that for any possible measurement of X, the absolute value of M must remain constant. The projective coordinate system allows covariant change of Y, i.e. change of units.
In practice, the unit system [X:1] is very rarely used, with Hz a more common variant, though less precise. To change units we have to solve the projective equation [1:1] = [9192631770:Y'], thus Y' = 1/9192631770. Now a measurement of X' in Hz has a representation [X':1/9192631770], whose absolute value is equal to another measurement in the [x:1] system.
It is clear that any number Y, Y', Y'' etc. is a dimensional unit, for example: 1 = 1 Caesium cycle / 9192631770 Hz, 1 = 1 KHz / 1000 Hz, etc. In teaching I would almost never mention the analogy to projective geometry, but always emphasize the point that changing units is equivalent to multiplication by unity, a dimensionless factor.
I plan to write all this up more clearly in the future. That could be a chance for me to look through your book or to think about corner cases, but as I said, I'm initially skeptical.
Cheers,
--Brad
[1] http://physics.princeton.edu/~mcdonald/examples/2cylinders.pdf Historical note: This problem was not "originally suggested" by me, it was "originally solved" by me. Kirk's first solution, which accompanied a journal rejection, was incorrect. I corrected him, and he published his subsequent work online. I have the emails. * * * They owe me a cracker at Princeton! * * *
On Thu, Jan 30, 2020 at 9:57 PM George Hart <george@georgehart.com> wrote:
I find that even people who work with dimensions all the time can have surprisingly different feelings about such questions and what the structure of the algebra of dimensions might be.
As with angle and dimensionlessness, torque and energy appear to have the have the same exponents in their dimensional terms (length^2 * mass^1 * time^-2 * charge^0) yet different physical roles in which they are never added, suggesting standard algebras of dimensions do not quite capture everything of interest even for defining something as basic as when addition of two quantities is defined.
These sorts of issues pop up in my book, but are not the central focus. As far as I know, no one has a fully satisfactory framework even for scalars. There is another level of richness when considering the dimensions of vectors, matrices, etc. But to the original point: including dimensions within the linear algebra brings out very clearly the difference between a vector and a co-vector.
George http://georgehart.com
While no expert in the area, I understand that co-vectors and indeed the whole idea of covariance and contrvariance is a topic usually seen at the point tensors are introduced, which is well beyond the point at which vectors are introduced in a typical Calculus III. As far as why the resistance to quaternions compared to standard vector analysis, there is an excellent history on just why the scientific community as a whole ended up using vectors that probably explains why we are where we are today. Check out “A History of Vector Analysis” by Michale J. Crowe. There is a cheap Dover edition available these days. A talk on the topic by the author can be found at https://www.researchgate.net/publication/244957729_A_History_of_Vector_Analy... Steve On Jan 29, 2020, at 10:46 AM, Tom Knight <tk@mit.edu<mailto:tk@mit.edu>> wrote: A huge disappointment from my MIT math and physics education was the lack of explanation early on of the difference between vectors and co-vectors. It wasn’t until I started reading about differential geometry that I realized the pervasiveness of this education-driven ignorance. All of a sudden lots of things that never made sense started clicking. How and why do we allow this fuzziness to continue? On Jan 28, 2020, at 11:16 PM, Fred Lunnon <fred.lunnon@gmail.com<mailto:fred.lunnon@gmail.com>> wrote: Good luck with convincing the engineers involved! Mind you, it is only getting on for 180 years ago that Hamilton came up with quaternions. So early days ... er, centuries? But perhaps it might be more constructive to attempt to understand why there is such instinctive resistance among engineers to the whole notion of quaternions, never mind more general geometric (Clifford) algebra. There's a frustrating pedagogical phenomenon involved in such investigations. While I can well recall my own sense of bewilderment on first encountering Hestenes' early book on these matters, I cannot muster the slightest insight into the cause of those conceptual difficulties. As a result, I now am stranded as far away from offering assistance to the uninitiated as I earlier was from receiving any: I simply cannot understand _why_ they can't understand. Anyway, here's a couple of possible clues: perhaps others can come up with more suggestions. (A) It's noteworthy that the first thing Heaviside did was to dissect Hamiton's elegant unity into "scalar" & "vector" parts, which went on to gain pretty much universal acceptance. There seems to be a mental hurdle in human minds obstructing the union of disparate familiar categories under a common umbrella: in this case, familiarity with angles & Cartesian coordinates actively obstructs conceptualisation of quaternions. (B) It is rarely made explicit that (like vectors) quaternions come in two flavours: polar & axial, depending on application. It seems that Hamilton himself was confused over this, which contributed to early controversy about their validity. There's a informative but slightly muddled paper on this topic somewhere on the internet which proceeds from the quaint premiss that they must exclusively be one or the other, despite the arguments put forward clearly illustrating a dichotomy. WFL On 1/28/20, Henry Baker <hbaker1@pipeline.com<mailto:hbaker1@pipeline.com>> wrote: Perhaps it's time to learn quaternions?? From comp.risks: Date: Fri, 10 Jan 2020 20:24:07 +0000 From: "Clive D.W. Feather" <clive@davros.org<mailto:clive@davros.org>> Subject: Boeing 737s can't land facing west (FAA) "The FAA received reports earlier this year of three incidents of display electronic unit (DEU) software errors on Model 737 NG airplanes flying into runway PABR in Barrow, Alaska. All six display units (DUs) blanked with a selected instrument approach to a runway with a 270-degree true heading, and all six DUs stayed blank until a different runway was selected. [...] The investigation revealed that the problem occurs when this combination of software is installed and a susceptible runway with a 270-degree true heading is selected for instrument approach. Not all runways with a 270-degree true heading are susceptible; only seven runways worldwide, as identified in this AD, have latitude and longitude values that cause the blanking behavior." (Note that this is all 6 displays on each plane, not 2 displays on each of three planes.) The runways in question are: Runway 26, Pine Bluffs, Wyoming, USA (82V) Runway 28, Wayne County, Ohio, USA (KBJJ) Runway 28, Chippewa County, Michigan, USA (KCIU) Runway 26, Cavern City, New Mexico, USA (KCNM) Runway 25, Barrow, Alaska, USA (PABR) Runway 28, La Mina, La Guajira, Colombia (SKLM) Runway 29, Cheddi Jagan, Georgetown, Guyana (SYCJ) (The numbers are magnetic bearings, whereas the problem is apparently related to true bearing.) Original FAA notice: <https://urldefense.proofpoint.com/v2/url?u=http-3A__rgl.faa.gov_Regulatory-5Fand-5FGuidance-5FLibrary_rgad.nsf_0_3948342a978cc27b862584dd005c1a60_-24FILE_2019-2D25-2D17.pdf&d=DwIGaQ&c=eLbWYnpnzycBCgmb7vCI4uqNEB9RSjOdn_5nBEmmeq0&r=vge6KOo90zMf7Wx14WFtiQ&m=d9KbpfFXHtZX1e5HYLaWDWd4OsOVzfm17SHlvvCQt-g&s=HSP37lAMGZkXVJYe097HUy8nRJ7oN406TYp56hEMkMU&e=> _______________________________________________ math-fun mailing list math-fun@mailman.xmission.com<mailto:math-fun@mailman.xmission.com> https://urldefense.proofpoint.com/v2/url?u=https-3A__mailman.xmission.com_cg... _______________________________________________ math-fun mailing list math-fun@mailman.xmission.com<mailto:math-fun@mailman.xmission.com> https://urldefense.proofpoint.com/v2/url?u=https-3A__mailman.xmission.com_cg... _______________________________________________ math-fun mailing list math-fun@mailman.xmission.com<mailto:math-fun@mailman.xmission.com> https://urldefense.proofpoint.com/v2/url?u=https-3A__mailman.xmission.com_cg...
participants (9)
-
Allan Wechsler -
Brad Klee -
Brent Meeker -
Fred Lunnon -
George Hart -
Henry Baker -
James Propp -
Lucas, Stephen K - lucassk -
Tom Knight