David K Butler

@davidkbutler

Maths Learning Support lecturer at Adelaide Uni (my views here). Grad Dip Ed & PhD Finite Geom. Love maths and helping people learn. he/him

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David K Butler@davidkbutler· 23d

#bloomscrolling

A single arum lily from above. It has a pure white spathe curled around a little yellow finger-shaped spadix. Broad tough dark green leaves are visible below.

David K Butler@davidkbutler· 24d

A combinatorics problem inspired by this little project my wife just finished: How many ways are there to arrange the drawers so that drawers with the same design aren’t next to each other vertically or horizontally?

A set of little wooden drawers three high by two wide. The fronts of the drawers have patterned paper on them. The paper on the top left and bottom right drawers has windows and flowers on a white background. The paper on the top right and middle left drawers is horizontal green and white stripes. The paper on the bottom left and middle right drawers is flowers on a green background.

David K Butler@davidkbutler· 28d

#bloomscrolling

A purple daisy. It has pale purple petals around d the outside, smaller darker purple petals further in and then even smaller even darker purple petals in the middle. Other flowers are visible in the background against dark green leaves.

David K Butler@davidkbutler· 28d

#bloomscrolling

In the background is a tangle of long fleshy spiky leaves with a ridge of spines along each side. In the foreground is a conical inflorescence of bright orange tube-shaped flowers. Only the lower flowers have opened.

David K Butler@davidkbutler· 28d

#bloomscrolling

On a grey twig, there are multiple puffs of red stamens growing directly out of fold lines in the bark. Narrow needle-like leaves also grow out of the twig at the right-hand end. (These are casuarina flowers, which I have never observed before today.)

David K Butler@davidkbutler· 29d

#bloomscrolling

A green field extends into the distance, dotted all over with bright yellow flowers. In the foreground several of the flowers are close enough to see. They are soursob flowers with their five round petals open to the viewer.

David K Butler@davidkbutler· 29d

#bloomscrolling

A dangling branch with several tiny gum blossoms. The blossoms are each a fluffy pink ring with a yellow cup in the centre. In the background are the long bright green leaves of the gum tree.

David K Butler@davidkbutler· Jul 13

Found this in my scrap paper during a meeting with my manager and had to suppress a guffaw.

A complicated geometrical construction on slightly crumpled paper with many straight lines in many directions, with several of the points labelled with small capital letters. Across one side of the paper is written in large capital letters fuuuck! (That is, fuck but with four u’s and a big exclamation mark.)

David K Butler@davidkbutler· Jul 9

#bloomscrolling

Several yellow flowers growing on long green stalks. They have a puff of orange stamens in the middle with many narrow overlapping petals surrounding it.

David K Butler@davidkbutler· Jul 9

These two triangles are *congruent*! Both have an edge of length c and a purple edge. They also both have an angle of 120°, though not between the two matching edges. But the angles that touch vertices are each 60° minus the same angle.

The labels x, y, z, w and v have been removed and two triangles have been roughly coloured in green. One triangle is enclosed by the top left edge of the big equilateral triangle, the top left edge of the yellow hexagon and the extended top edge of the yellow hexagon. The other is enclosed by the bottom left edge of the big equilateral triangle and the two main diagonals of the yellow hexagon.

David K Butler@davidkbutler· Jul 9

Ooh I’ve seen another pair of similar triangles. I suppose whenever you have two meeting chords of a circle you’d be guaranteed to make two pairs of similar triangles like this, hey?

Two triangles have been roughly coloured in green from the previous picture. They are between the downwards-sloping main diagonal of the yellow hexagon and the top left edge of the big equilateral triangle.

David K Butler@davidkbutler· Jul 8

I’m going to relabel everything and I reckon similar triangles/ lines cut by parallel lines will relate things in pairs.

The two parts of the bottom left edge of the big equilateral triangle are labelled x and y from bottom to top. The three parts of the top left edge of the big equilateral triangle have been labelled z, w, v from left to right. Various edges have been labelled c and b and 2a.

David K Butler@davidkbutler· Jul 8

I’ve just seen another line I can draw that may yield fruit.

All labels and colouring-in have been removed and a horizontal pink line has been drawn cutting the yellow hexagon in half.

David K Butler@davidkbutler· Jul 8

Those first two similar triangles I noticed have two matching edges that are edges of the yellow and pink hexagons. If I label the longest edge of the smaller triangle x then the longest edge of the bigger triangle is x/b*c, because /b*c is how you get from one triangle to the other.

The two triangles on either side of the bottom left edge of the purple equilateral triangle have been shaded green again. The longest edge of the smaller triangle is labelled x and the longest edge of the other triangle has been labelled x divided by b times c.

David K Butler@davidkbutler· Jul 8

So let me give the edges of the hexagons names. a for the small orange one, b for the middle-sized pink one and c for the big yellow one. Because regular hexagons can be cut into equilateral triangles, their main diagonals are twice their edges.

No green triangles now. But labels have been added to various edges. Three edges of the yellow hexagon are labelled lowercase c. Two edges of the pink hexagon are labelled lowercase b. One edge of the orange hexagon is labelled lowercase a, while the horizontal main diagonal is labelled 2a. The already-drawn main diagonal of the yellow hexagon is labelled 2c.

David K Butler@davidkbutler· Jul 8

And here’s another pair. I know they’re similar because of the vertically opposite angles and one of the other angles is 60°.

A different pair of triangles has been roughly highlighted in green. The same one as last time, and the one the leftmost corner of the equilateral triangle cut off by the main diagonal of the yellow hexagon.

David K Butler@davidkbutler· Jul 8

And here’s another pair. They’re similar because of the parallel edges created by the edge and main diagonal of the yellow hexagon.

A triangle has been roughly highlighted in green. It’s enclosed by the top edge of the purple equilateral triangle, the main diagonal of the yellow hexagon and the top edge of the yellow hexagon.

David K Butler@davidkbutler· Jul 8

Here’s one pair. I know they’re similar because they have a pair of vertically opposite angles and one of the other angles is definitely 120°.

Two triangles have been highlighted roughly in green. They are on either side of the lower left edge of the purple equilateral triangle. The bigger one fills out the bottom left corner of the yellow hexagon and the smaller one fills out the space between the yellow hexagon and the pink hexagon.

David K Butler@davidkbutler· Jul 8

Wait if I draw in the other side of that triangle it has to be equilateral because the leftmost corner is subtended by the opposite edge and I know it subtends an angle of 60°. How satisfying. So satisfying I could stop there and be ok with it if I never came back.

The ends of the two precious purple lines have been joined with a third purple line completing an equilateral triangle.

David K Butler@davidkbutler· Jul 8

Hmm these two chords both subtend an angle of 60° at the circumference and so they have to be the same length. Don’t care if it’s helpful because it’s pretty cool.

Two purple chords of the circle have been added. One goes from the leftmost corner of the yellow hexagon to the bottom left corner of the pink hexagon. The other goes from the rightmost corner of the orange hexagon to the bottom left corner of the pink hexagon.

David K Butler@davidkbutler· Jul 8

And another one from that same corner of the yellow hexagon along the top edge and through the middle of the orange hexagon.

A green line has been added from the top left corner of the yellow hexagon along the top edge and through the middle of the orange hexagon to meet the circle at the rightmost corner of the orange hexagon.

David K Butler@davidkbutler· Jul 8

Ooh there’s one here from one corner of the yellow hexagon through the opposite corner and along the edge of the pink hexagon.

A blue line has been drawn from the top left corner of the yellow hexagon through the opposite corner and along the edge of the pink hexagon to the circle again.

David K Butler@davidkbutler· Jul 8

First, what do I see? Check the ALT text for what I’m seeing.

There’s a big red circle. A yellow regular hexagon is oriented with top and bottom edges horizontal. The leftmost point of the top horizontal edge and the next corner of the hexagon anticlockwise are on the circle. A pink regular hexagon has an edge along the opposite edge of the yellow hexagon, with one shared corner. This corner is the right hand end of the yellow hexagon’s bottom horizontal edge. The next corner of the pink hexagon anticlockwise (that is, the leftmost end of its bottom horizontal edge) is also on the circle. An orange hexagon shares one corner with the big yellow hexagon. It’s the rightmost end of the top horizontal edge of the yellow hexagon and the leftmost corner of the orange hexagon. The sloping edge of the orange hexagon clockwise from that corner lies along the edge of the yellow hexagon. The opposite corner of the orange hexagon lies on the circle. The orange hexagon has a big 6 written inside it and the pink hexagon has a question mark written inside it.

David K Butler@davidkbutler· Jul 7

Me with a couple of kids looking at a deltahedron.

Me (a white man in jeans and a t-shirt) holding up a shape made of plastic triangles that click together for two young boys in school polo tops to look at. On the table in front of us are several more shapes also made of coloured triangles.

David K Butler@davidkbutler· Jul 6

David K Butler@davidkbutler· Jul 3

#MathSky Eighth and final post on two-sided ruler construction is done! I've saved my favourites for last. In this one I show how to construct an equilateral triangle and a regular pentagon, and I give myself some stars.

Several lines in various colours drawn on paper. Some of the points have been labelled with letters. Part of the construction is a big five-pointed star, which has been coloured in roughly in orange.

David K Butler@davidkbutler· Jul 3

#MathSky The penultimate post on two-sided ruler constructions is done! This one is about finding where circles meet without actually drawing the circles. davidkbutler.xyz/2026/07/03/t...

A very complicated diagram drawn with various coloured pencils on paper. There are points labelled A, B, R, S, P, Q, X, G and H, and a lot of parallel horizontal lines, as well as some parallel lines in other directions and even more lines at other angles.

David K Butler@davidkbutler· Jul 1

The next blog post in the series about two-sided ruler constructions is finally done! This one is about constructing parallel lines. davidkbutler.xyz/2026/07/02/t...

A page with many lines in multiple directions in multiple colours and various points labelled with letters has a ruler lined up with two points with one hand holding it steady and the other drawing along the edge of the ruler with a pink pencil.

David K Butler@davidkbutler· Jun 28

#MathSky Fifth blog post on two-sided ruler constructions is now complete! This one is on constructions that make perpendicular lines. davidkbutler.xyz/2026/06/28/t...

A construction is drawn on a piece of paper with several lines in various colours, and some of the points are labelled with letters. A clear ruler has been aligned to two of the labelled points and is being held in place by a hand, while another hand draws along the side of the ruler with a pencil.

David K Butler@davidkbutler· Jun 24

#bloomscrolling