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sun-jester.bsky.social
SunJester
@sun-jester.bsky.social
48 followers 37 following 160 posts
He/him. Not very focused. Likes math, science & art. Go eclecticism!

I also write a little bit...
for short stories and poems see https://www.wattpad.com/user/Sun_Jester
for longer stories https://www.royalroad.com/profile/686795
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sun-jester.bsky.social
For some reason the additive formula for binomial coefficients is often proven using the formulas for the binomial coefficients, even though there is a way better combinatorial proof.

Just remember that n choose k is how many ways there are to choose k balls out of n.

#math #mathematics #mathsky
Combinatorial proof of the addition formula for binomial coefficients
October 6, 2025 at 8:27 PM
sun-jester.bsky.social
#AugustOfGroups: every day of August I'll post all groups whose order is the number of that day. Today August 30th.

There are four groups of order 30: Z/30Z, D15, (Z/5Z) × D3 and (Z/3Z) × D5.

I'm a particular fan of the symmetry in those last two

#math #maths #mathematics #mathsky
(Z/5Z) × D3 is the group of rotational symmetries of a pentagon times all symmetries of a triangle. (Z/3Z) × D5 is the group of rotational symmetries of a triangle times all symmetries of a pentagon. In other words: in the above picture you're allowed to transform one in any (symmetric) way you want, but can only rotate the other.
August 30, 2025 at 7:05 PM
sun-jester.bsky.social
#AugustOfGroups: every day of August I'll post all groups whose order is the number of that day. Today August 18th.

There are 5 groups of order 18: Z/18Z, (Z/3Z) × (Z/6Z), (Z/3Z) × S3, D9, (Z/3Z)³ ⋊ (Z/2Z)

The latter has a nice matrix representation

#math #maths #mathematics #mathsky
Z/3Z ⋊ Z/2Z where Z/2Z acts via inversion, is the only truly new group. IIt has this nice matrix representation.
August 18, 2025 at 8:39 PM
sun-jester.bsky.social
Finally, the Pauli group is the group consisting of the Pauli matrices, multiplied by ±1 and ±i. As the name implies, this group is useful in quantum mechanics (6/6)

#math #maths #mathematics #mathsky
The Pauli matrices sigma_x, sigma_y and sigma_z. These are used to describe spin in (non-relativistic) quantum mechanics.
August 17, 2025 at 12:29 PM
sun-jester.bsky.social
Then there are two more interesting ones:

(Z/4Z) ⋉ (Z/4Z) and (Z/4Z) ⋉ V4.

Whereas in G × H (he product), the groups G and H don't really interact, in G ⋉ H (the semidirect product) they do (see image). Both of the top pairs only have a single non-trivial semidirect product. (3/6)
In the usual product group, G and H don't really interact. In the semidirect product, there is some homomorphism phi from H to Aut(G) which is used to make the groups 'link together' in a way. Given two groups G, H there are usually many phi's to choose from, but for the two pairs at the top you can either take phi trivial (everything gets sent to the neutral element) or choose a single non-trivial one. Excercise: what non-trivial automorphism phi is used?
August 17, 2025 at 12:29 PM
sun-jester.bsky.social
The dicyclic group Dic3 consists of the three roots of unity in the complex plane together with the quaternion j, and then all possible products

More generally, Dicn is the group generated by the nth roots of unity and the quaternion j. (3/3)

#math #maths #mathematics #mathsky
The three cube roots of unity (blue) on the complex plane (grey) together with the quaternion j. Note that the full group would require 4 dimensions to draw
August 12, 2025 at 7:06 PM
sun-jester.bsky.social
Given a permutation s of {1, 2, ..., n}, we have an inversion if i < j, but s(j) < s(i). If the amount of inversions of s is even, it's called an 'even permutation'. An is the group of even permutations.
Beyond that: A4 is also the group of rotational symmetries of a reguler tetrahedron (2/3)
A regular tetrahedron can be rotated along several axes, like the two shown here. One passes through the midpoints of two edges, the other through a vertex and the midpoint of the opposing side
August 12, 2025 at 7:06 PM
sun-jester.bsky.social
#AugustOfGroups: every day of August I'll post all groups whose order is the number of that day. Today August 9th.

There are 2 of order 9: Z/9Z and (Z/3Z)²

(Z/3Z)² is the rotational symmetries of a pair of equilateral triangles (each turns on its own)

#math #maths #mathematics #mathsky
(Z/3Z)² is the group of rotational symmetries of a pair of equilateral triangles. (1,0) corresponds to rotating the yellow one 120° to the right, (2,0) rotates the yellow one 120° to the left. (0,1) rotates the right 120° to the right and (0,2) to the left. For example: (1,2) means 'rotate the yellow triangle 120° to the right and the green one 120° to the left' Z/9Z is the group of rotational symmetries of a regular nonagon
August 9, 2025 at 4:31 PM
sun-jester.bsky.social
The first two are nothing new. (Z/2Z)³ is the symmetry of three pairs of points, D4 is the group of symmetries of a square.

Q is the group of the quaternions i,j and k (hence the name). The last one has geomertic interpretations, but these are quite hard (2/2)

#math #maths #mathematics #mathsky
Quaternion group is the symmetry group of this highly compliacted object (see https://archive.bridgesmathart.org/2014/bridges2014-143.pdf) The defining relationship for the quaternion group Q.
August 8, 2025 at 6:06 PM
sun-jester.bsky.social
#AugustOfGroups: every day of August I'll post all groups whose order is the number of that day. Today August 8th, the first actually interesting day.

There are 5 groups of order 8: Z/8Z, the group (Z/2Z) × (Z/4Z), (Z/2Z)³, the dihedral group D4 and the quaternion group Q.
(1/2)
(Z/2Z)³ is the symmetry group of these six points: (1,0,0) swaps the two blue ones, (0,1,0) the red ones and (0,0,1) the orange ones. D4 is the symmetry group of a square. Note that there are permutations of the green, blue, yellow and purple point that do NOT appear. This shows that D4 and S4 are distinct!
August 8, 2025 at 6:06 PM
sun-jester.bsky.social
S3 is the symmetry group of permutations of three elements.
D3 is the symmetry group of an equilateral triangle. Amazingly enough, these two are the same!
Symbolically, S3 ≅ D3 (2/2)

#math #maths #mathematics #mathsky
Cayley table of S3 and D3. Note that we could give r^2, rm and r^2m distinct names if we wanted to, but that would only make things more complex. This group is completely defined by asserting that r^3 = e, m^2 =e and mr = r^2m. All other identities follow (this is an example of a 'group presentation') The group of symmetries of an equilateral triangle (the colors are only there so you can see the difference). r corresponds to a 120° degree rotation to the right, m to a reflection across the vertical line through the center. Note that every single permutation of the blue, green and yellow dots appears, this is a visual proof that S3 ≅ D3.
August 6, 2025 at 8:32 PM
sun-jester.bsky.social
#AugustOfGroups: every day of August I'll post all groups whose order is the number of that day. Today August 6th.

There are two groups of order 6: Z/6Z and S3 the symmetric group of 3 elements (our first non-abelian group!)

#math #maths #mathematics #mathsky (1/2)
Cayley table of Z/6Z.
August 6, 2025 at 8:32 PM
sun-jester.bsky.social
#AugustOfGroups: every day of August I'll post all groups whose order is the number of that day. Today August 5th.

There is again only 1 group of order 5: Z/5Z, the group of rotational symmetries of a pentagon (By now the pattern in Z/nZ is clear)

#math #maths #mathematics #mathsky
Cayley table of Z/5Z Z/5Z gives the group of rotational symmetries of a regular pentagon, 1 rotates to the right by 72°, 2 to the right by 144°, 3 to the left by 144° and 4 to the left by 72°
August 5, 2025 at 9:17 PM
sun-jester.bsky.social
Second: the Klein four-group V, our first non-cyclic group!

V is also (Z/2Z)² and is the group of symmetries of a (non-square) rectangle. (2/2)

#math #maths #mathematics #mathsky
Cayley table of the Klein four group The Klein four group is the symetry group of a rectangle, a is a reflection across the vertical, b across the horizontal and c is a 180° rotation. Now a^2 = b^2 = c^2 = e and the product of any two is the third.
August 4, 2025 at 4:11 PM
sun-jester.bsky.social
#AugustOfGroups: every day of August I'll post all groups whose order is the number of that day. Today August 4th, the first interesting day.

There are two groups of order four:

first Z/4Z (the integers modulo 4). This is the group of rotational symmetries of a square (1/2)
Cayley table of Z/4Z Z/4Z is the group of rotational symmetries of a square, 1 is a 90° rotation to the right, 2 is a 180° rotation and 3 is a 90° rotation to the left, so 1 + 1 = 2, 1 + 2 = 3, 1 + 3 = 0 2 + 2 = 0, 2 + 3 = 1 3 + 3 = 2
August 4, 2025 at 4:11 PM
sun-jester.bsky.social
#AugustOfGroups 3: every day of August I'll post all groups with order the number of that day

there is only one group of order 3 (I swear it gets more interesting later on): Z/3Z, the integers modulo 3
Visually: the group of rotational symmetries of a triangle
#math #maths #mathematics #mathsky
Cayley table of Z/3Z Z/3Z as group of rotational symmetries of an equilateral triangle: 1 corresponds to turning 120° to the right, 2 corresponds to a 120° turn to the left. Rotating 120° to the right twice is the same as rotating 120° to the left, rotating 120° to the right thrice is the same as doing nothing. In other words: 1 + 1 = 2, 1 + 1 + 1 = 0
August 3, 2025 at 7:42 PM
sun-jester.bsky.social
The Cayley table of the trivial group is a bit silly, but here it is anyway
Cayley table of trivial group
August 2, 2025 at 5:23 PM
sun-jester.bsky.social
#AugustOfGroups: every day of August I'll post all groups whose order is the number of that day. August 2nd.

there is only one group of order 2: Z/2Z, the integers modulo 2

#math #maths #mathematics #mathsky
Cayley table of Z/2Z
August 2, 2025 at 5:21 PM
sun-jester.bsky.social
The determinant is often seen as a rather strange thing to define, the formula coming seemingly out of nowhere. In fact, it can be derived by simply asserting it has the following three properties:

#math #mathematics #mathsky #matrices #determinant
Property 1: swapping two columns of a matrix switches the sign of the determinant. Property 2: the determinant, when seen as a function on the columns, is multilinear. Property 3: the determinant of the identity matrix is 1. This is just a normalisation.
July 24, 2025 at 3:07 PM
sun-jester.bsky.social
~Poem #26 ~

All Bells Can Ding.
Even Flying Ghosts Hear.
I Jest,
Knowledge Leaves More No’s.
Only People Question Rights, Sometimes.
Time Used Voraciously, Wastefully.
Xericicity Yields Zealotry.

#poem #poetry #writingcommunity #writing

For more poems, please check out A Bridge Between Two Cs
Cover of A Bridge Between Two Cs
June 30, 2025 at 4:47 PM
sun-jester.bsky.social
I thought it'd be nice to have a #poem, where the roles of the title and the actual texts have switched

#writing #WritingCommunity #poetry

If you like it, you can read some more of my poems in my collection [A Bridge Between Two Cs](www.wattpad.com/story/393302...)
Poem titled ~To prepare the reader. Let them know what they’re in for, before the poem starts~ What's a Title For?
June 19, 2025 at 8:58 PM
sun-jester.bsky.social
Cauchy's theorem is easily one of my favourite results from #math (why do you think it's my banner image?), it's almost unbelievable:

The integral of a derivable function over a closed curve in the complex plane is always* zero!

*under some (weak) conditions

#mathsky #mathematics
Cauchy's theorem, drawn in sand of the Sahara
June 14, 2025 at 2:44 PM
sun-jester.bsky.social
Another #poem I wrote.

~Rauchabzug~

#poetry, #writing #WritingCommunity
The alarm is going off WAAAH, WAAH! I go outside, it hurts my ears it hurts my head WAAAH, WAAAH! the sound, incessant sound probably just burnt toast WAAAAH, WAAAAH! I press the code to make it stop silence, finally … smoke’s coming into the room, filling it up can’t see a thing a red box on the wall, button behind plastic it says Rauchabzug
June 13, 2025 at 3:47 PM
sun-jester.bsky.social
A simple #animation I made in #geogebra to illustrate the trigonometric functions. Sine is blue, cosine red, tangent orange and cotangent green

#math #mathsky #mathematics
June 12, 2025 at 7:45 PM
sun-jester.bsky.social
It is one of my favourite results from #mathematics and used in so many situations it's absurd:

The Cauchy–Schwarz inequality,
(also discovered by Bunyakovsky)
its beauty and versatility
is matched by hardly anything you’ll ever ever see.

#math #MathSky
The Cauchy-Schwarz inequality, the dot product of two vectors is never greater than the product of the norms of the vectors
June 7, 2025 at 3:26 PM