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Unique Fractions between 0 and 1, visualized in polar coordinates (2018).

I love this so much so I'm sharing it again.

Each fraction is a pair of numbers k/n, where k is a totient of n.
The denominator n is mapped to the radius n and the number k/n is mapped to the angle 2pi*k/n.

I conceptualized and created this with R.

Proof in a toot:

If a² = a for a ∈ R, then a² - a = 0. So, a(a-1) = 0. So a(a-1) ∈ I since any ideal of R contains the zero of R. Then since I is a prime ideal, we know a ∈ I or a-1 ∈ I.
Finally, if a ∈ I, then a + I = I. Thus (a - 1) + I = -1 + I. This is not an element of I as long as I is properly contained in R. Thus, R/I is a two-element set {0 + I, 1 + I}. Then |R/I| = 2.

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Let R be commutative and unital with a² = a for all a ∈ R. If I is a prime ideal, then |R/I| = 2.

I don't want anyone to solve this for me, but am I on the right track saying that every r + I ∈ R/I is either 1 + I or I itself?

Real analysis 2: anyone have an example of a weird metric that isn’t l1, lp, sup norm, or discrete?

Abstract algebra 2 progress: when he hit us with “R[X] / <x^2 + 1> is isomorphic to C” the class went “wooooaaahhh”

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Sneak peek at my latest project - guess what the pattern is?

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Looking for friends to do napkin math with over coffee.

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Ladies, if he:

-never responds to your texts
-goes out of his way to make you feel bad
-multiplies rapidly at temperatures between 68 and 104 degrees

He's not your man. He's a foodborne pathogen.

It’s hard to shake the male expectation of being “tough” and “alpha” but you have to realize, alpha males are unhappy. They are insecure. They don’t know what it’s like to express feelings with the heart. Love is gonna save the world. Yes I struggle with depression but the path away from this is through acceptance and vulnerability

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position: -1.120504275365011 + 0.22661320403256938i
pixel width: 9.763494047536691e-06

I’ve had such a turnaround in my life this past couple months. All you need is to send a little love to yourself. I’ve been beating myself up so hard for so long, I forgot what it was like to feel confident no matter the circumstances. I am happy to say I’ve had a week free of anxiety attacks. I cried a few times this week, but you gotta let yourself cry right?

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@birdman We need to get our cities working on lowering light pollution, banning all synthetic pesticides, and filling our cities to the brim with weeds, ponds, and flowers.

I want, more than anything, a way for citizens to participate. But the participation we need is financial. The vast majority of carbon pollution comes from the richest ten percent of the population. Everyone needs to vote with their wallets and put these fuckers out of business.

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So, insect populations have dropped drastically since the 1970s. We may be seeing a lot of insect-pollinated plants going extinct.

We need to tell people why they have to care about this.

The issue is, nobody responds to fear that is this catastrophic. They just shut down. We have to manage our fear-based language and replace it with statements of love and optimism for the Earth.

“Mathematicians are usually categorized as the ‘Euler type’ or the ‘Gauss type’.”

I feel like I’m categorized as the poor sucker standing outside the window looking on. Javert style

Found a great comment on Reddit today.

“Finitely generated nilpotent-free k-algebras are to varieties as rings are to schemes.”

So, curves are associated with friendly linear-algebra-like objects but if you do the same with a ring it ends up looking not like a curve at all?

Well, I just got a job offer, but i've still got dreams of research. Someday I'll be there.

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