Pinned toot

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.

I think it surprises people that I'm an optimist.

Yes, I am very honest about societal issues, but in the grand scheme of American history, we've never been in a better position for significant social change.

The work is often hard and thankless, but when you look at what we've overcome to get to this point, I think there are very real reasons to hope.

Throw

A Hamiltonian cycle on the vertices of the 120-cell.

Source code and more explanation: http://community.wolfram.com/groups/-/m/t/1273027 https://mathstodon.xyz/media/Kzas7Me2PkF5Dztcl_M

@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.

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.

Found a great comment on Reddit today. #algebraicgeometry #algebra

“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?

Mathematics at U of AZ. Interested in algebra and art.

Joined Jun 2017