#ALevelMaths In my previous video I looked at how to work out the equation for the envelope of a family of curves. This time I look at some situations here that method comes up with something that isn't an envelope. Enjoy!

#AlevelMaths Envelopes appear in Further Maths, at least for Edexcel and OCR/MEI, and the formula for an envelope is a bit mysterious, so I've been digging into it, and have made a video you might enjoy. youtu.be/AneMSG1EO1U

@davidkbutler.bsky.social It is a magnificent Autumn for berries here in the UK. Just look at this spindle bush in a field near my home.

#MathsToday #AlevelMaths The Edexcel spec talks about 1-1 or many-1 functions and 1-many mappings, but doesn't mention many-many mappings. One of my pupils reports that his teacher says "you can't have many to many mappings". Well you certainly can, though perhaps not in an Edexcel exam.

I'm exploring envelopes, in the mathematical sense, which have always puzzled me. It's not a corner of mathematics I've hade reasons to study before. If f(t,x,y)=0 is a family of curves, t being a parameter, I follow the proof that we obtain an equation for the envelope ...

The brewers of Proper Job beer have produced a strong IPA called Big Job. When I was toddler, doing a big job was a euphemusm for having a crap. These things change rapidly, so presunably the brewery employs nobody young enough to remember this. Should I tell them?

I've published my latest video, on the history of the metre, focussing on the aspects of the history that as a mathematician I find particularly interesting. For example, do you know which unit is now defined to be an irrational multiple of a metre? youtu.be/jsW8H76nBKo?...

Gosh, I've been slow. I've only just realised that the fact that 100 kilometres is almost exactly 54 nautical miles is not a coincidence. A nm was originally 1 arc minute around a great circle on the Earth, and a km was originally 1/10000 of the distance from the North pole to the equator via Paris.

There was a recent thread on pedantry in mark schemes, which prompts a thought on pedantry in mathematics. In brief there are two sorts, one good, one a pain. A friend says mathematicians are creative pedants, ...

The hedgrows round here are festooned with red haws, blue sloes and blackberries to an extent I don't remember ever seeing before.

Do you remember the rod, pole or perch, and do you wonder where they come from? Ever wondered where the strange factor of 11 comes from in the number of yards in a mile. All is revealed in my latest video: youtu.be/6uaBJq-gd_o

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That was a close game!

Is there a standard set of colours to use that will work for people with various forms of colour blindness? For use, eg, when I'm making a video, when of course I don't know who will watch. I have in mind something that says "If you need five contrasting colours, here are the ones to use".

I suspect most pea plants are simply connected. Not this one. Cue other mathematically interesting plant pics.

Hurrah!

This supermarket trolley wheel misunderstood the instructions concerning which axis it is supposed to rotate about.

Did you know that the inradius, and all three exradii, of a Pythagorean triangle are all integers? Here is a video exploring this delightful fact via Heron's formula for the area of a triangle and Euclid's formula for generating primitive pythagorean triples. youtu.be/Zoib28QEDq4

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Next year I am going to do some maths olympiad coaching, which I am really looking forward to. Does anyone have any books or other resources to recommend, apart from past papers of course.

There is something evocative (of what I'm not sure) about the light of a summer's evening. I guess it has to do with the prolonged twilight. This year, I noticed it for the first time this evening, walking home from the pub.

The new pope has a maths degree, and is clearly an intellectual heavyweight. I googled "mathematician pope", and it offered me Sylvester II who pontificated 999 to 1003. Fascinating guy, and like Leo, quite the intellectual. I wonder if Robert Prevost considered calling himself Sylvester V.

Pray tell me, Dr Fermi, how many flowers do you think there are on my wisteria.

This is fun.

Granddaughter has always loved rainbows. We made a rainbow-themed picnic for her 12th birthday. Note the black things at the end. They represent ultraviolet!

Our fritillaries are very happy by the pond.

#MathsToday #AlevelMaths Fruitful discussion prompted by two flawed questions. One says "... equilibrium ... find the coefficient of friction". It doesn't say limiting friction or anything to imply that, so actually we can only find a lower bound on μ.

There's another one: QEA, quod est absurdum, to be used at the of a proof by contradiction. Have you ever met that?

#MathsToday #ALevelMaths My student tackled an integration problem that ended up with a cubic to be solved. She has a new calcualtor, and hasn't yet worked out how to use it to solve cubics, so she used Newton-Raphson. Hurrah!

#MathsToday Newton-Raphson. I love it. I often complain about daft questions in the text book, but thus is great. Kepler's equation. Though it doesn't mention Kepler by name, it says "an astronmer" and "a planet" with eccentricity 0.1. That's Mars. It is often said that Kepler showed that. ..

#MathsToday Newton Raphson. I love it. I often complain about Qs in the text book, but this is brilliant. Kepler's equation. It talks about "an astronomer", and "a planet" with eccentricuty 0.1. That's Mars. It is often said the Kepler showed that the planets move in ellipctical orbits.

My electric car has a dial labelled consumption, in miles per kilwatt-hour, instantaneous and long-term average. That's a strange unit, but it has the dimensions of force^(-1), so it must really be showing the driving force! So let's convert my long term average, 4.1, to Newtons. ...

Mathematics in School in shreds. Does anyone want a dog?

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#MathsToday Spent some time disentangling my poor S's brain that had got all twisted up over the two meaning of mean - the expected value of a distribution and the sum of some observations divided by their number.

It's my birthday. My age in years is a Lucas number, and the name of four places in the US. Do we have any places in the UK whose names are numbers?

A short #ALevelMaths thread on yesterday's #MathsToday post. Take the parametric curve f(x,y)=0, and consider the curves f(x,y)=λ. Τhese are the projections onto the (x,y) plane of the level curves of the surface z=f(x,y), ιe the intersection of that surface with the plane z=λ.

#MathsToday Today's stupid question, find the tangent to a curve given implicitly at a given point. Fair enough you might think, except that the given point is not on the given curve. Interesting discussion about what happens in this situation. Level curves.

#MathsonHoliday We visited a local market to see signs saying things like "3 × 10". It turns out that means 3 items for 10 euros. More like 10 ÷ 3, I'd say

I am reading about Schrödinger's eqn in 1D. The author says d²Ψ/dx² is the curvature of Ψ, which it isn't according to the mathematical defn of curvature. So do physicists use the word curvature differently, or is there a tacit assumption that dΨ/dx is small, or is this author just wrong?

#MathOnHoliday Who thought of having the numbered lift buttons arranged horizontally? And worse than that, right to left! To be fair, there are two lifts, mirror images, so the other one had its buttons left to right, as all right-thinking horizontally-buttoned lifts should.

#MathsOnHoliday I'm rather enjoying watching this rotating door. Its circumference has 2 closed thirds and 2 open sixths so there can bo no draughts, (it is windy here). Also it has IR sensors, so stops if no-one wants to go through. Simple pleasures. Oh, and the breakfast is excellent!

#MathsToday We thought about a 3-D printer printing a semicircular cylinder, and constructed the path of the nozzle as a 3-D parametric curve. Also we constructed a 2-D parametric curve in geogebra and also a point moving along the curve with a slider. All this is treating the parameter as time.

#MathsToday My 67-year old student who is studying with me because he wants to to understand more is seriously keen. I am unavailable next week, so he has asked for an extra lesson this week.

#MathsToday, well yesterday. I was taught QED, quod erat demonstrandum (which was to be proved), at the age of about 12, when we were taught Eulcid-style proofs. TIL QEF, quod erat faciendum (... done), QEI, quod erat inveniendum (... found.) Has anyone every seen these? Were they really used?

#Mathstoday #AlevelMaths What I call the generalized quotient rule. If y = u/v^k, y' = (vu'-kuv')/v^(k+1). I think it is rather beautiful, and sure saves a lot of writing and prevents a lot of mistakes, particularly if k=½. I think it is worth learning in its own right. Do you teach it?

What is the oldest consumable in your household? I bought this tub of grease about 59 years ago. I don't use it often but used a little yesterday morning. Should last a few more years.

#MathsToday parametric equations. Has anyone got any real world applications that are not contrived?

@moudhy.bsky.social ‪ We know the symbols the Babylonians used to write numbers, but they must surely also have been able to say the numbers out loud, so the numbers must had pronounceable names (as we have twenty-six as well as 26). Do we know these names? Were they ever written down?

#UKMathsChat Working in the garden this morning, I have found setting π = 3 to be perfectly satisfactory. I was making a cylinder of chicken wire to protect a young tree from muntjac.