Mark Thornber
@drmthornber.bsky.social
Retired A level maths teacher, current examiner. Enjoy maths puzzles and cryptic crosswords.
All readable. Pick the one where you have the least background:
The Mathematical Theory of The Top Felix Klein
Prime Numbers and the Riemann Hypothesis Barry Mazur
Random Walks and Electric Networks Doyle & Snell
The Mathematics of Medical Imaging Timothy Feeman
The Mathematical Theory of The Top Felix Klein
Prime Numbers and the Riemann Hypothesis Barry Mazur
Random Walks and Electric Networks Doyle & Snell
The Mathematics of Medical Imaging Timothy Feeman
October 16, 2025 at 10:09 PM
All readable. Pick the one where you have the least background:
The Mathematical Theory of The Top Felix Klein
Prime Numbers and the Riemann Hypothesis Barry Mazur
Random Walks and Electric Networks Doyle & Snell
The Mathematics of Medical Imaging Timothy Feeman
The Mathematical Theory of The Top Felix Klein
Prime Numbers and the Riemann Hypothesis Barry Mazur
Random Walks and Electric Networks Doyle & Snell
The Mathematics of Medical Imaging Timothy Feeman
The group of unit quaternions (isomorphic to the matrix group SU(2)) is used extensively in modern 3D computer graphics. The sort used in all those games your students play when they should be doing FM homework.
October 6, 2025 at 7:06 PM
The group of unit quaternions (isomorphic to the matrix group SU(2)) is used extensively in modern 3D computer graphics. The sort used in all those games your students play when they should be doing FM homework.
#FridayFive
1. Tom Dooley - Doc Watson
2. I Don’t Like Mondays - The Boomtown Rats
3. Jean - Oliver
4. Lucky - Britney Spears
5. Cherry Bomb - The Runaways
1. Tom Dooley - Doc Watson
2. I Don’t Like Mondays - The Boomtown Rats
3. Jean - Oliver
4. Lucky - Britney Spears
5. Cherry Bomb - The Runaways
September 12, 2025 at 8:56 PM
#FridayFive
1. Tom Dooley - Doc Watson
2. I Don’t Like Mondays - The Boomtown Rats
3. Jean - Oliver
4. Lucky - Britney Spears
5. Cherry Bomb - The Runaways
1. Tom Dooley - Doc Watson
2. I Don’t Like Mondays - The Boomtown Rats
3. Jean - Oliver
4. Lucky - Britney Spears
5. Cherry Bomb - The Runaways
Autocorrect seems to have added 0.000977… !
September 12, 2025 at 5:24 PM
Autocorrect seems to have added 0.000977… !
To solve:
First note eq3/eq1 gives 1/32=α^2r^3 so we do need the positive root after all!
Now eq2 divided by this gives
35/4=(1+r+2r^2+r^3+r^4)/r^2
This rearranges to a quartic and the factor theorem quickly gives r=2 etc.
First note eq3/eq1 gives 1/32=α^2r^3 so we do need the positive root after all!
Now eq2 divided by this gives
35/4=(1+r+2r^2+r^3+r^4)/r^2
This rearranges to a quartic and the factor theorem quickly gives r=2 etc.
September 12, 2025 at 3:51 PM
To solve:
First note eq3/eq1 gives 1/32=α^2r^3 so we do need the positive root after all!
Now eq2 divided by this gives
35/4=(1+r+2r^2+r^3+r^4)/r^2
This rearranges to a quartic and the factor theorem quickly gives r=2 etc.
First note eq3/eq1 gives 1/32=α^2r^3 so we do need the positive root after all!
Now eq2 divided by this gives
35/4=(1+r+2r^2+r^3+r^4)/r^2
This rearranges to a quartic and the factor theorem quickly gives r=2 etc.
The second equation is wrong. It needs α^2.
The fourth equation should be 1/1024=0.000977α^4r^6
(Is it obvious that we need the positive root?)
The fourth equation should be 1/1024=0.000977α^4r^6
(Is it obvious that we need the positive root?)
September 12, 2025 at 3:45 PM
The second equation is wrong. It needs α^2.
The fourth equation should be 1/1024=0.000977α^4r^6
(Is it obvious that we need the positive root?)
The fourth equation should be 1/1024=0.000977α^4r^6
(Is it obvious that we need the positive root?)
Ford Econoline -Nanci Griffith
By The Time I Get To Phoenix - Glen Campbell
Piss Factory - Patti Smith
Arthur’s Theme - Christopher Cross
I’ll Follow The Sun - The Beatles
#FridayFive
By The Time I Get To Phoenix - Glen Campbell
Piss Factory - Patti Smith
Arthur’s Theme - Christopher Cross
I’ll Follow The Sun - The Beatles
#FridayFive
September 5, 2025 at 3:24 PM
Ford Econoline -Nanci Griffith
By The Time I Get To Phoenix - Glen Campbell
Piss Factory - Patti Smith
Arthur’s Theme - Christopher Cross
I’ll Follow The Sun - The Beatles
#FridayFive
By The Time I Get To Phoenix - Glen Campbell
Piss Factory - Patti Smith
Arthur’s Theme - Christopher Cross
I’ll Follow The Sun - The Beatles
#FridayFive
It can! Consider f(x,y)=ax^2+2bxy+cy^2.
fxx=2a fxy=2b and fyy=2c, so all values are possible.
This has a sta. pt. at the origin. The sign of b^2-ac tells you if z=f(x,y) is a parabolic or hyperboloid.
fxx=2a fxy=2b and fyy=2c, so all values are possible.
This has a sta. pt. at the origin. The sign of b^2-ac tells you if z=f(x,y) is a parabolic or hyperboloid.
August 28, 2025 at 4:44 PM
It can! Consider f(x,y)=ax^2+2bxy+cy^2.
fxx=2a fxy=2b and fyy=2c, so all values are possible.
This has a sta. pt. at the origin. The sign of b^2-ac tells you if z=f(x,y) is a parabolic or hyperboloid.
fxx=2a fxy=2b and fyy=2c, so all values are possible.
This has a sta. pt. at the origin. The sign of b^2-ac tells you if z=f(x,y) is a parabolic or hyperboloid.
“Sing epsilon delta, Stewart and Tall” (Google it!)
June 22, 2025 at 6:39 PM
“Sing epsilon delta, Stewart and Tall” (Google it!)
Graham Cumming posted a long thread on X today, with his best guesses as to what went wrong and why things turned out as badly as they did. I suspect he is right on all counts!
June 20, 2025 at 8:21 PM
Graham Cumming posted a long thread on X today, with his best guesses as to what went wrong and why things turned out as badly as they did. I suspect he is right on all counts!
#FridayFive
1. Long Hot Summer - Tom Robinson Band
2. I Am A Town - Mary Chapin Carpenter
3. Here Comes the Sun - The Beatles
4. If You Can’t Stand The Heat - Bucks Fizz
5. Summer Breeze - The Isley Brothers
1. Long Hot Summer - Tom Robinson Band
2. I Am A Town - Mary Chapin Carpenter
3. Here Comes the Sun - The Beatles
4. If You Can’t Stand The Heat - Bucks Fizz
5. Summer Breeze - The Isley Brothers
June 20, 2025 at 8:05 PM
#FridayFive
1. Long Hot Summer - Tom Robinson Band
2. I Am A Town - Mary Chapin Carpenter
3. Here Comes the Sun - The Beatles
4. If You Can’t Stand The Heat - Bucks Fizz
5. Summer Breeze - The Isley Brothers
1. Long Hot Summer - Tom Robinson Band
2. I Am A Town - Mary Chapin Carpenter
3. Here Comes the Sun - The Beatles
4. If You Can’t Stand The Heat - Bucks Fizz
5. Summer Breeze - The Isley Brothers
OCR follows Rudin etc.
Item 8.01c on the specification states:
‘“Divergence” can refer to sequences that are bounded or unbounded.’
Item 8.01c on the specification states:
‘“Divergence” can refer to sequences that are bounded or unbounded.’
June 20, 2025 at 7:01 PM
OCR follows Rudin etc.
Item 8.01c on the specification states:
‘“Divergence” can refer to sequences that are bounded or unbounded.’
Item 8.01c on the specification states:
‘“Divergence” can refer to sequences that are bounded or unbounded.’
I don’t think this is correct. Here is professor Peter Coles (a Newcastle lad) with the original Latin and the translations.
https://telescoper.blog/2019/11/13/newtons-laws-in-translation/
https://telescoper.blog/2019/11/13/newtons-laws-in-translation/
June 17, 2025 at 5:45 PM
I don’t think this is correct. Here is professor Peter Coles (a Newcastle lad) with the original Latin and the translations.
https://telescoper.blog/2019/11/13/newtons-laws-in-translation/
https://telescoper.blog/2019/11/13/newtons-laws-in-translation/
This list worked for me
https://bhi.co.uk/repairer/
https://bhi.co.uk/repairer/
June 15, 2025 at 4:09 PM
This list worked for me
https://bhi.co.uk/repairer/
https://bhi.co.uk/repairer/
Are cointerior angles really equal?
June 6, 2025 at 10:22 PM
Are cointerior angles really equal?
For polynomials, tangent lines can be found by looking for repeated roots. No need for limits.
We find m so that p(x)-p(a)=m(x-a) has a repeated root.
This gives m=p’(a)
For quadratics it’s good old b^2-4ac=0
We find m so that p(x)-p(a)=m(x-a) has a repeated root.
This gives m=p’(a)
For quadratics it’s good old b^2-4ac=0
May 3, 2025 at 6:19 PM
For polynomials, tangent lines can be found by looking for repeated roots. No need for limits.
We find m so that p(x)-p(a)=m(x-a) has a repeated root.
This gives m=p’(a)
For quadratics it’s good old b^2-4ac=0
We find m so that p(x)-p(a)=m(x-a) has a repeated root.
This gives m=p’(a)
For quadratics it’s good old b^2-4ac=0
It’s definitely a US thing. Just check any of their doorstep-sized textbooks.
I always thought it was easier to remember than concave/convex, but confusion arises later when convex is used to mean “the curve is under the chord”
I always thought it was easier to remember than concave/convex, but confusion arises later when convex is used to mean “the curve is under the chord”
February 27, 2025 at 5:31 PM
It’s definitely a US thing. Just check any of their doorstep-sized textbooks.
I always thought it was easier to remember than concave/convex, but confusion arises later when convex is used to mean “the curve is under the chord”
I always thought it was easier to remember than concave/convex, but confusion arises later when convex is used to mean “the curve is under the chord”
There are a number of similar triangles. The ratios of corresponding lengths will be equal.
February 19, 2025 at 8:28 PM
There are a number of similar triangles. The ratios of corresponding lengths will be equal.
See my previous reply.
February 16, 2025 at 9:00 AM
See my previous reply.
Lists are arrays. You can access individual items with indexing. For example L1[2]
February 13, 2025 at 11:48 PM
Lists are arrays. You can access individual items with indexing. For example L1[2]
I usually buy direct from the Double Two website. Lots of colours available
https://www.doubletwo.co.uk/mauve-long-sleeve-non-iron-cotton-rich-shirt/
https://www.doubletwo.co.uk/mauve-long-sleeve-non-iron-cotton-rich-shirt/
January 26, 2025 at 7:08 PM
I usually buy direct from the Double Two website. Lots of colours available
https://www.doubletwo.co.uk/mauve-long-sleeve-non-iron-cotton-rich-shirt/
https://www.doubletwo.co.uk/mauve-long-sleeve-non-iron-cotton-rich-shirt/
The first 3 terms are (z-3)^2+1. Try the substitution u=z-3, followed by completing the square.
January 26, 2025 at 2:02 PM
The first 3 terms are (z-3)^2+1. Try the substitution u=z-3, followed by completing the square.
You should try the Guardian quick cryptic. Each one uses 4 different clue types and they give examples of each.
https://www.theguardian.com/crosswords/series/quick-cryptic
https://www.theguardian.com/crosswords/series/quick-cryptic
Quick cryptic | The Guardian
Latest news, sport, business, comment, analysis and reviews from the Guardian, the world's leading liberal voice
www.theguardian.com
January 11, 2025 at 6:12 PM
You should try the Guardian quick cryptic. Each one uses 4 different clue types and they give examples of each.
https://www.theguardian.com/crosswords/series/quick-cryptic
https://www.theguardian.com/crosswords/series/quick-cryptic