David K Butler
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davidkbutler.bsky.social
David K Butler
@davidkbutler.bsky.social
1.8K followers 650 following 5.7K posts
Lecturer at Maths Learning Centre, Uni Adelaide (my views here). Grad Dip Ed & PhD Finite Geom. Love maths and helping people learn. he/him
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davidkbutler.bsky.social
One more collection of flowers.
Large clusters of bright orange flowers consisting of hundreds of vertical spikes. A cluster of white butterfly-tree blossoms, with more out of focus in the background. Each flower has five petals, with the top one having two brushes of pink stripes, several curved white stigmas and a bright pink stamen. Clusters of roses in various shades of pink. The more tightly furled ones are in darker shades of pink. Between large crinkly-edges leaves grow tall spikes with white flowers, each showing its mouth under a purple sepal. One spike in the foreground is in bright sunlight while the ones in the background are in shadow.
November 11, 2025 at 6:47 AM
davidkbutler.bsky.social
#read2025

“Father Brown” (the Wordsworth Classics edition) by GK Chesterton.

A collection of Father Brown stories written between 1911 and 1936. Not *all* of the stories, but a decent number of them. I’d never read them before and asked my father in law if I could borrow his.
The cover of my father-in-law’s copy of Father Brown (the Wordsworth Classics edition) by GK Chesterton. It shows a painting of a London street in the fog.
November 9, 2025 at 7:44 AM
davidkbutler.bsky.social
That area can be cut into a triangle and a sector. The triangle has base x and height √[1-x^2] so its area is 1/2 x √[1-x^2]. The sector area would be found based on an angle β. And actually sin β=x, so β=arcsin(x). So the sector has area 1/2 arcsin(x).
Same picture as before but now a diagonal line is drawn from the origin to the top of the vertical line at x making a triangle. The bottom edge is labelled x, and the vertical edge is labelled the square root of 1 minus x squared. The angle between the diagonal line and the y-axis is labelled beta. A horizontal dotted line goes along the top making a new triangle. That dotted line is also labelled x.
November 4, 2025 at 7:24 PM
davidkbutler.bsky.social
Here’s a picture.
Very rough sketch of a pair of axes with a semicircle. The horizontal axis is labelled t and the vertical axis is unlabelled. A position on the right-hand half of the t-axis is labelled x, and a vertical line goes from there up to the semicircle. The part of the semicircle between this line and the vertical axis is shaded.
November 4, 2025 at 7:14 PM
davidkbutler.bsky.social
Oliver decided the best pillow was a magazine.
An orange and white cat sleeps with his white paws over his face and his back legs apart. He lies on a white bedsheet, bur with his head on an open magazine.
November 2, 2025 at 3:00 AM
davidkbutler.bsky.social
Me at our university’s veterinary school with some animal skeletons.
A smiling man in a blue t-shirt (me) in front of a large animal skeleton. It has big splayed toes on its front legs and a long snout looking down. (It’s a tapir skeleton, if you want to know.) A smiling man in a blue t-shirt (me) next to an animal skeleton in a classroom. The skeleton has a big shell and a small head on a long neck. (It’s a turtle skeleton.)
October 30, 2025 at 4:49 AM
davidkbutler.bsky.social
A new collection of flowers.
Several tobacco flowers growing on a long curving stalk. The flowers are very long skinny tubes with a slightly fatter part near the end. They are mostly yellow, except the tips of the ones still in the process of opening are green. A spray of white pointy flower buds on spindly stalks, with exactly one flower open. This flower has six white petals, three rounded and three pointy with purple tips. In the centre are several fuzzy white and yellow stamens. Fuzzy in the background is the facade of the Art Gallery of South Australia. Two bright red-orange pomegranate blossoms. Each has several crinkly petals around a large shock of yellow stamens. Further up on the branch are a couple of waxy bright red-orange unopened flower buds. A delicate pink bottlebrush blossom with dark green leaves growing out of its end.
October 28, 2025 at 5:52 AM
davidkbutler.bsky.social
This I posted on Twitter in 2020 is currently experiencing a resurgence.
A screenshot of a Twitter post. It’s from David Butler (me) at 4:55 16/7/2020. The text says: Separating rational functions using partial fractions is a change of basis. I knew this already. But today I realised that this means you can do it WITH A MATRIX. The image shows maths: To separate 3 x + 1 over x+1 times x+2 into partial fractions … 1 over x+1 times x+2 = 1 over x+1 plus -1 over x+2 x over x+1 times x+2 = -1 over x+1 plus 2 over x+2 2 by 2 matrix with first column 1 and -1 and second column -1 and 2 times column vector 1 and 3 = column vector -2 and 5 Therefore 3x+1 over x+1 times x+2 = -2 over x+1 + 5 over x+2
October 25, 2025 at 10:19 PM
davidkbutler.bsky.social
Right. Here’s everything I said so far:
Equations written on paper. k + b = v a + d = v s + t = v m = 0 d = -v g - t = X g - d = b Two columns of letter values. v = 1 d=-1 g=-4 a=2 b=-3 v=-1 d=1 g=4 a=-2 b=3
October 25, 2025 at 6:43 AM
davidkbutler.bsky.social
Here they are.
So they’d be 1/√2(±1±i).
A diagram of the unit circle. Two diagonal lines have been drawn in, with points marked in green on the unit circle where the diagonal lines meet it.
October 20, 2025 at 10:00 PM
davidkbutler.bsky.social
I could also have done a trig substitution I reckon.
x^2+4 is similar in spirit to finding the long side of a right-angled triangle using Pythagoras’s theorem.
I’ll call the angle opposite x by u.
A right-angled triangle with horizontal short side labelled 2, and vertical short side labelled x, and hypotenuse labelled square root of x squared plus 4. The angle opposite x is labeled u.
October 20, 2025 at 9:44 PM
davidkbutler.bsky.social
Yet another collection of flowers.
A cluster of purple flowers. Each has two big purple petals and a tube-like structure with a spray of white stamens at the end. Some flowers have the two petals closed together over the structure like a tent. In one of these little tents, a bee is nuzzling. A bright poppy with four big orange petals fringed with yellow. It’s surrounded by other poppies in various states of unfurl. The whole photo is filled with little white flowers with green-yellow centres growing on grey-green stems. Each flower has five heart-shaped white petals. Salvation Jane / Patterson’s Curse blossoms. Each is pinky purple making a loose shallow trumpet.
October 18, 2025 at 1:47 AM
davidkbutler.bsky.social
Another collection of flowers.
A branch with tiny bright green leaves and many little pink five-petalled flowers. A flower with three pink petals, growing from a hanging plant with red-purple leaves, each with two green stripes along their length. Several pink flowers among bright green untidy leaves. Each flower has five petals that are pale pink at the edges. The two petals on the sides are dark pink nearer to the centre. The two petals as the top and the one at the bottom are yellow-orange near the centre with many brown spots. Pink apple blossoms, and some spherical unopened buds.
October 17, 2025 at 12:41 AM
davidkbutler.bsky.social
Flowers still continuing.
Among grey-green leaves are many white flowers with yellow centres. Each flower seems to be a decagon with ten triangular sectors. One flower has a bee nuzzling its head in the yellow centre. Several bright orange flowers. Each flower has many pointed petals arranged around a circle. The part of each petal near the middle circle is dark, forming a brown border to the orange centre circle. Several white daisies with many long narrow petals arranged around bright yellow centres. Among green leaves are a multitude of purple flowers with white centres. Each flower is a pentagon made of five triangular segments.
October 16, 2025 at 6:33 AM
davidkbutler.bsky.social
Flowers continue yet again.
Three trumpet-shaped yellow flowers with five petals surrounding the opening, growing at the end of a branch with textured green leaves. A huge rose with peach petals near the centre and pale yellow petals near the edge, with the outermost petals fringed in pink. A few unopened buds nestle in the green leaves around it. A big protea blossom with about a hundred yellow curved structures ending in rounded tips. More blossoms are visible out of focus in the background. A spray of flowers in various shades of pink and variable number of petals. A bee nuzzles in the centre of one of the flowers.
October 16, 2025 at 6:19 AM
davidkbutler.bsky.social
Another subtractions-only game of Digimoji (Digit Disguises with emojis).
Fruit and Vegetables Emoji Calculations apple minus banana = apple grapes minus apple = cross corn minus grapes = cross mushroom minus corn = cross eggplant minus mushroom = avocado avocado minus mushroom = orange orange minus mushroom = apple carrot minus mushroom = corn watermelon minus mushroom = carrot
October 16, 2025 at 6:09 AM
davidkbutler.bsky.social
GNU Tabitha Brown 2014 - 2025
Tabitha Brown’s face. She is a tabby cat with one green eye and one brown eye and a pink nose. She has slightly orange fur on her nose and white fur on her chin. She is being cradled in my arm wearing a blue and white striped jumper. She looks up and to the right with sparkling eyes.
October 14, 2025 at 3:29 PM
davidkbutler.bsky.social
I had a go using only subtractions. These ten results were enough to figure which digit 0 to 9 had which emoji disguise.
Fruit and Vegetables Emoji calculations Avocado minus watermelon = cross Watermelon minus carrot = watermelon Watermelon minus orange = mushroom Avocado minus eggplant = cross Eggplant minus corn = cross Corn minus grapes = watermelon Corn minus banana = orange Corn minus apple = mushroom Banana minus orange = cross Watermelon minus grapes = grapes
October 12, 2025 at 6:47 AM
davidkbutler.bsky.social
Flowers continue even more.
Two bright yellow irises touching each other. Each has six petals in pairs top and bottom forming a little tunnel between them. The bottom petals are very big and have black and brown stripes leading into the tunnel. Between the tree tunnel pairs, tiny pale yellow petals curve upwards. A third flower hasn’t opened yet and is bright yellow spike. The leaves in the background are long and straight and pointy. Further below is a tangle of old dead leaves. Several little flowers each with five teardrop-shaped petals that only touch in the very centre where there are several very short yellow-tipped stamens. The flowers range from solid warm yellow to a pale yellow that’s almost white. They grow from stems that also have many shiny green leaves. A red brick wall is in the background. A single bright red flower hangs downwards from a stem with several oval-shaped leaves except for a sharp tip. The flower is a funnel shape made of five overlapping petals. Two little white flowers grow among green leaves with yellow splotches. The flowers have five petals that curl behind with white stamens and a yellow stigma in the middle.
October 12, 2025 at 12:25 AM
davidkbutler.bsky.social
Oh, and because I am who I am, here is a graph:
A scatterplot. Title: What I noticed along the way. Observation length across 2025. Horizontal axis is labelled with dates 1/10/2024, 1/11/2024, ... , 1/09/2025, and a vertical gridline is visible at each of these dates. Vertical axis is labelled Number of characters, and a horizontal gridline is drawn at the numbers 0, 50, 100, ..., 350. Blue dots scatter across the graph between and on the gridlines, ranging between about 60 and about 300.
October 9, 2025 at 9:45 PM
davidkbutler.bsky.social
Flowers continue more.
Tight clusters of many many cream-coloured flowers, each with five round separated petals, spiky stamens and a ring of yellow in the centre. A bee sits on the edge of one cluster. Clusters of pink and white flowers. Each has some pink petals facing up and some pink petals facing down with some curved white petals facing down in the middle. A bright yellow flower among green leaves. The flower is made of a couple of layers of yellow petals with short faint orange stripes near the centre. Two clusters of tiny flowers, several yellow flowers surrounded by pink flowers, all with orange centres. They grow from a branch with dark green textured leaves.
October 9, 2025 at 9:30 PM
davidkbutler.bsky.social
Flowers continue again.
Two brilliant magenta flowers, each with five long pointy petals curling back, and several pink stamens with white tips and a white stigma curling up. In the background are green butterfly-shaped leaves. Two clusters of gum blossoms, one cluster tight and the other loose. Each has a red-orange cup and a multitude of pale yellow stamens. A billy blossom: a perfect globe of tiny bright yellow flowers on a long grey-green stem. In the background are a tangle of stringy grey-green leaves and several blossoms that aren’t quite ready yet. Many lavender blossoms, with two in focus in the centre of the photo. They have an egg-shaped collection of little grey-purple flowers with several big purple petals at the top. A bee sits on one of the blossoms in focus.
October 9, 2025 at 9:24 PM
davidkbutler.bsky.social
Flowers continue.
A very tall spike filled with small flowers, each with blue petals and stringy white stamens. A bee has alighted near the bottom of the spike. Straight green leaves and more flower spikes are visible in the background. Lemon tree blossoms, each with five white waxy elongated oval petals and a cluster of yellow-tipped stamens in the centre. They are surrounded by green leaves dotted with more flowers and pink unopened buds. A bright red-orange rose. Glorious red fluffy bottlebrush blossoms hang above a grey chickenwire fence.
October 6, 2025 at 10:03 PM
davidkbutler.bsky.social
Still more flowers again.
Clusters of little flowers each with five warm yellow petals curved like parts of spheres. Against bright green round leaves, two flowers with five petals surrounding a funnel fuzzy with stamens. One flower’s petals are yellow, while the other’s are pale orange with darker orange stripes towards the centre. Bell-shaped bright blue flowers hanging from tall green stems rising from between long green leaves, all in front of a stone wall. A metre-long inflorescence hangs downwards, displaying hundreds of hand-sized bright red flowers.
October 4, 2025 at 3:17 AM
davidkbutler.bsky.social
This came up in the staff meeting today: how the ordinary English meaning of the word “independent” does not describe at all how a Venn diagram looks, and what to do to fix that. #MathSky www.adelaide.edu.au/mathslearnin...
A collection of nine diagrams. Each is a ten-by-ten grid with a four-by-five rectangle in its bottom left corner shaded with wavy lines. Each also has a five-by-six rectangle shaded in grey, overlapping with the wavy-shaded rectangle by a different amount in each diagram. They each have a label underneath too. In the first diagram, the grey rectangle covers the wavy-shaded rectangle completely and the label says “Not independent”. In the second diagram, the two rectangles overlap on 15 squares and the label says “Not independent”. In the third diagram, the two rectangles overlap on 12 squares and the label says “Not independent”. In the fourth diagram, the two rectangles overlap on 8 squares and the label says “Not independent”. In the fifth diagram, the two rectangles overlap on 6 squares and the label says “Independent!” in capital letters. In the sixth diagram, the two rectangles overlap on 4 squares and the label says “Not independent”. In the seventh diagram, the two rectangles overlap on 2 squares and the label says “Not independent”. In the eighth diagram, the two rectangles overlap on 1 squares and the label says “Not independent”. In the ninth diagram, the two rectangles do not overlap and the label says “Not independent”.
September 30, 2025 at 8:52 AM