#roots of unity

[mathhombre]

Cyclotomic Roots

Mathober Day 21: Cyclotomic

Either new to me or forgotten, I loved digging into this idea. The nth cyclotomic polynomial is the largest degree factor of x^n-1 which divides no x^m-1 for m<n. You have to put it that way, because if a prime p divides n, then the pth cyclotomic divides x^n-1. Anyhow, the result of this is that the roots of the nth cyclotomic correlate to the fractions i/n that are irreducible.

So for x^15-1, we get 1/15, 2/15, 4/15, 7/15, 8/15, 11/15, 13/15, 14/15 as the roots of unity of the 15th cyclotomic, x^8-x^7+x^5-x^4+x^3-x+1. You can find them by dividing by the prime factors cyclotomics, which all have the form x^(p-1)+x^(p-2)+...+x^2+x+1.

Just beautiful! I added the sectors to make it prettier to see. Not sorry.

I couldn't get GGB to multiply out the roots of unity to get the polynomials, but that would be pretty cool. Play with it as is, though, here.

Oh, ok. One more: 99

[ayxza]

Tree topper ? Nah use the roots of unity

[bibliophilea]

Ooh, I love the Gaussian integers! Gotta be one of my favorite rings! I had forgotten that about prime numbers in Z: that they're prime in Z[i] iff their remainder when divided by 4 is 3. That's a fun little proof, too!

I personally am a big fan of roots of unity - love that there are so many complex numbers out there where, when you multiply them to themselves enough, you get back to 1! And they all sit on the unit circle! And they act exactly like Q in R from 0 to 1 - and that means they're dense on the unit circle, but there's always something between any two roots of unity that's on the unit circle but isn't itself a root of unity! And because that something is on the unit circle, it stays on the unit circle when you multiply it to itself - but it never reaches 1. It just goes on forever and ever, filling in infinitely many (but not all of) the holes left by the roots of unity on the unit circle! So cool!

One thing about me is I infodump about math subjects on request

[blackstarlineage]

Types of Black People to Avoid From a Garveyite Perspective

Marcus Garvey’s teachings emphasize Pan-Africanism, self-reliance, and pride in African heritage. From his perspective, certain mindsets and behaviours among Black people hinder the collective progress of the race. This post outlines the types of individuals Garvey warned against and why their actions conflict with his philosophy.

1. The Self-Hating Negro

Those who lack pride in their African identity or seek to align with oppressive systems instead of embracing their heritage.

2. The Betrayer or Opportunist

People who align with colonial or imperial powers for personal gain, selling out their community in the process.

3. The Idle and Unproductive

Garvey emphasized hard work and self-reliance. Those who perpetuate laziness or refuse to contribute to the community weaken.

4. The Divisive

Individuals who sow discord or prioritize personal grievances over unity.

5. The Apologist for Oppression

Those who excuse or justify racism, colonialism, or exploitation instead of resisting it.

6. The Lackey or "House Negro"

People who serve as tools of oppression within the Black community, often working to uphold systems of exploitation.

7. The Disconnected

Those who neglect the struggles of their brothers and sisters across the African diaspora.

8. The Cynic or Defeatist

Individuals who spread hopelessness and believe liberation is impossible, stifling collective progress.

9. The Assimilationist

Those who reject African culture to assimilate into white-dominated societies, often seeking validation from oppressors.

10. The Colonial-Minded Intellectual

Educated individuals who use their knowledge to support colonial ideologies rather than fight for Black liberation.

11. The Non-Believer in Black Institutions

People who doubt the ability of Black people to build and sustain their own schools, businesses, and organizations.

12. The Religious Misleader

Religious figures who preach passivity and submission rather than encouraging active resistance and empowerment.

13. The Consumer Over Creator

Those who consume from exploitative systems but fail to support Black-owned businesses or initiatives.

14. The Envy-Driven

Individuals who sabotage others out of jealousy rather than celebrate their success as a win for the community.

15. The Perpetual Victim

Garvey believed in resilience and empowerment, not adopting a victim mindset that leads to inaction.

16. The Uninformed or Ignorant

People who refuse to educate themselves on African history, culture, or the realities of systemic oppression.

17. The Political Sellout

Politicians or leaders who prioritize personal power over the well-being of their people.

18. The Violent Toward Their Own

Garvey emphasized unity. Those who harm or exploit their own people weaken the community.

19. The Materialistic

Individuals who prioritize wealth and status over collective liberation.

20. The Fearful or Complacent

Those who fear challenging systems of oppression or accept the status quo.

21. The "Divide and Conquer" Advocate

People who promote tribalism, colourism, or internal divisions that undermine unity.

22. The Global Isolationist

Garvey believed in global solidarity. Ignoring the struggles of Africans worldwide weakens the movement.

23. The Culturally Oblivious

Individuals who neglect the preservation of African culture, language, and traditions.

24. The Misguided Activist

Those who harm liberation movements through counterproductive or extreme methods.

25. The Short-Term Thinker

People who focus on immediate gains without considering the long-term impact on future generations.

26. The Hypercritical Without Action

Critics who offer no solutions or refuse to contribute to progress.

27. The Neglectful Parent

Parents who fail to instill pride, education, and self-reliance in their children.

28. The Exploiter or Disrespecter of Black Women

Garvey emphasized the importance of both men and women in building a strong and self-reliant community. Those who exploit or disrespect Black women cause deep harm to the entire movement.

29. The Perpetrator and Promoter of Negative Stereotypes of Black Men and Women

Those who perpetuate negative stereotypes harm the collective image and self-esteem of Black people.

Examples of Harmful Stereotypes:

Stereotypes About Black Men: Promoting images of Black men as inherently violent, hypersexual, lazy, or criminal. These narratives contribute to societal discrimination and the dehumanization of Black men.

Stereotypes About Black Women: Reinforcing depictions of Black women as angry, overbearing, promiscuous, or unworthy of respect. These stereotypes undermine the dignity of Black women and lead to real-world harm.

Why This Conflicts with Garveyism:

Garvey called for the upliftment of the entire race, which includes challenging racist caricatures and media portrayals. Promoting stereotypes reinforces systemic oppression, fosters self-hatred, and fractures community unity.

"We must canonize our own heroes, create our own martyrs, and glorify our own symbols," Garvey said, urging Black people to celebrate their greatness instead of internalizing harmful narratives.

30. The Doubter of Pan-Africanism

Rejecting the idea of a unified global African identity weakens collective strength.

31. The Glorifier of Interracial Relationships Over Their Own Race

Individuals who prioritize dating outside their race, often due to internalized racism or disdain for their own community, undermine racial solidarity and family structures.

32. The Denier of African Heritage

Those who reject their African roots or refuse to identify as part of the African diaspora, fostering division and rejecting the foundation of Pan-Africanism.

33. The Disrespecter of Black Men and Boys

Individuals who belittle, demean, or fail to support Black men and boys contribute to the weakening of the community.

34. The Uninformed or Ignorant

Garvey strongly believed in education and the power of knowledge. Black people who remain ignorant about their history, culture, and the systems that oppress them harm the movement.

Why This Conflicts with Garveyism:

Ignorance weakens collective strength by promoting misunderstanding, inaction, and misplaced priorities. Education is a tool for liberation, and those who refuse to seek knowledge risk falling prey to manipulation and exploitation.

35. The Believer in Pseudo Subjects and History

Garvey valued accurate knowledge and warned against false ideologies or distorted versions of history.

Examples of Pseudo Subjects:

False Histories: Believing in fabricated or distorted versions of African history that lack historical evidence.

Pseudo-Sciences: Embracing unscientific or baseless theories about health, biology, or spirituality.

Why This Conflicts with Garveyism:

Pseudo subjects distract from the real work of liberation, confuse the community, and undermine credibility. Garvey’s philosophy calls for rooting oneself in truth and using knowledge as a foundation for empowerment

In Garvey's Words:

"A people without the knowledge of their past history, origin, and culture is like a tree without roots."

Garvey’s philosophy urges us to avoid behaviors and attitudes that undermine the pride, unity, and collective progress of Black people globally. Embracing our heritage, building strong communities, and fostering solidarity across the diaspora are key to liberation.

[blaithnne]

[sefaradweb]

Tiffany Haddish en el evento “Stand Together” en apoyo a Israel

🇪🇸 El 12 de noviembre de 2024, Tiffany Haddish participó en el evento "Stand Together" en Washington, D.C., donde miles de personas se reunieron para mostrar su solidaridad con Israel. Durante su emotivo discurso, Tiffany reflexionó sobre su propio descubrimiento de raíces judías, expresando su apoyo en un momento marcado por el dolor y la unidad de la comunidad judía. Mencionó las luchas actuales, como los 101 rehenes en Gaza, el aumento del antisemitismo, y el odio enfrentado en algunos campus universitarios. Haddish resaltó que la unidad no se basa en estar de acuerdo en todo, sino en apoyarse mutuamente en los momentos críticos. Con su mensaje inclusivo, destacó la diversidad en la comunidad judía y la importancia de la solidaridad ante las dificultades actuales.

🇺🇸 On November 12, 2024, Tiffany Haddish attended the "Stand Together" event in Washington, D.C., where thousands gathered to express their solidarity with Israel. In her heartfelt speech, Tiffany reflected on discovering her own Jewish heritage and voiced support during a time of grief and unity within the Jewish community. She spoke of current struggles, such as the 101 hostages in Gaza, rising anti-Semitism, and hatred encountered on college campuses. Haddish emphasized that unity is not about agreeing on everything but standing together in critical times. With her inclusive message, she celebrated the diversity within the Jewish community and stressed the importance of solidarity in the face of today’s challenges.

[elohim-israel]

Christianity is attempting to teach humans oneness and unity. People caught up in skin complexions and nationalism can not comprehend that.

Yes I. That’s a deep and true word.

Christianity, at its core, is about the oneness of humanity in Christ—“There is neither Jew nor Greek, slave nor free, male nor female, for you are all one in Christ Jesus” (Galatians 3:28). The Gospel calls us beyond divisions of tribe, race, and nation into a higher unity, rooted in the Spirit of God.

Tewahedo (ተዋሕዶ, täwaḥədo) is a Ge’ez word that means “being made one” or “unity.”

“In the mystic tradition of the Church, there is a unity of faith and doctrine. The unity of the Church, however, must be manifested in the unity of its members. Unity implies the kind of love that Christ showed to us on the Cross. Unity implies that all Christians are one in the spirit of humility and forgiveness.” — H.I.M. Haile Selassie I, Address to the World Council of Churches, Addis Ababa, 1971

💚💛❤️

[civilization-illstayrighthere]

guys guuuys so i live on a very narrow street in the city of Baltimore between two large high-traffic streets. It’s the smallest a street can be to not be considered an alley: it’s one way, cars can’t park on the sides, and kinda dark. Last year we got ZERO trick or treaters. I had to sit on the corner of one of the bigger streets to give things out. But this year i was determined. I took out a chair, put on my warmest flannel bat costume, stuck satchel in a doggie costume and sat outside. One little old lady came out to give satchel treats. “I don’t think we’ll get many this year.” She said. I agreed but stayed out.

And uhm. 🥹 a neighbor came out from across the street and introduced himself. He just moved in. He was happy to see me sitting out cause he wasn’t sure if kids would knock (and they generally don’t, tbh.) so he also set up a table and chair and sat outside.

And then another neighbor came out.

And then another.

Guys we were flagging down groups of kids walking by. Yelling “WE GOT CANDY! WE GOT TRICKS AND TREATS!” on our block six-seven entire houses came out to sit on their stoops and give out candy! We got TONS of kids! And people talked and got to know each other!! Because i sat outside first!!!!!

And at the end of the night, when everyone was going inside and pulling in their chairs, the new neighbor told me thanks for sitting outside, that I started it all. And like. Waaaughjghghgbg. I’m so happy.

[peninsularian]

1977, Abe Broadway posse again

[goldieclaws]

WIP of a batch of sprites I'm making for a new Visual Novel (because I love adding new projects to my already very long list) set in my Blue Bones universe. I haven't started writing the script yet, but I do have detailed notes and, as you can see here, I'm currently working on the art for it.

I might randomly switch between this story and my remake of AVIS., if only because I was getting frustrated with the code of my two Unity projects and I'd like to get this story out of my head sooner rather than later (with breaks in between of course).

All of the art will be uploaded to my Patreon in due time, with Ronin's concepts and sprites already uploaded and Calliope's scheduled for tomorrow.

(Ronin [left] and Flynn [middle] use he/him, Calliope [right] uses she/her)

Reblogs appreciated, thank you! 💖✨

Patreon | Tw//tter | AO3 | Itchio | Commissions | Webcomic

[9misoundsystem]

l próximo 𝟭𝟴 𝗱𝗲 𝗡𝗼𝘃𝗶𝗲𝗺𝗯𝗿𝗲 se reúnen los colectivos de soundsystem de la comunidad en un acto de solidaridad y unidad. Bajo el nombre "Unity is Strength," este evento tiene el propósito de recaudar fondos para enviar ayuda humanitaria a Palestina a través de asociaciones comprometidas como BDS País Valencià, UNRWA y CAPP.

La situación en Palestina es urgente y requiere de la colaboración de todos nosotros para brindar apoyo a quienes más lo necesitan. Es por eso que la aportación íntegra de todas las entradas vendidas será donada a estas destacadas asociaciones. Cada euro recaudado se convertirá en un rayo de esperanza para las comunidades que enfrentan desafíos diarios en Palestina.

El precio de la entrada es de tan solo 10€. Incluso si no puedes asistir al evento, puedes comprar tu entrada para colaborar y contribuir a la ayuda humanitaria que tanto se necesita.

[arctic-hands]

So is Nintendo going to rise to the occasion and be the unlikely hero or...?

[sacredmoonbeamtarot]

Forever the hermit with a magic core.

It’s important to know your home within. 🪐⚛️

[dlgarcia5]

FORGOTTEN

[l3gaci-1926]

Indigenous Black Americans, we know who we are!

[linuxgamenews]

Discover the New Adventures in Roots of Pacha

1.3 update Sun and Moon unleashes for the stone-age farming sim game Roots of Pacha on Steam Deck, Linux, Mac, and Windows PC. Thanks to the creative minds at Soda Den for bringing us further content. Which you can find via Steam and Humble Store with its 96% Very Positive reviews. If you’ve been looking for an excuse to return to Roots of Pacha or try it out for the first time—congrats, the stars have aligned. The cozy Stone Age farming sim just released its massive 1.3 update, and it’s packed with fresh content that’ll keep you busy for hours. Think new lands, new characters, new mechanics, with a whole lot more prehistoric goodness on Steam Deck and Linux. This 1.3 update, called Sun and Moon, is the biggest one yet—and it’s here to shake up your mid and late game like never before. So, what’s in this bad boy? First off, two whole new regions to explore: the wild Mograni Mountains and the lush Yakuan Islands. Each of them has three unique sub-areas filled with stuff to discover, including new rituals, jaw-dropping sights, and cultures with their own vibes. You’ll even get to join their festivals, like playing dice with the Mograni or jamming out with the musical Yakuans. Not just a vacation—this is full-on cultural immersion, Stone Age style.

Roots of Pacha: Sun and Moon 1.3 Update Trailer

New mechanics? Oh yeah. Ever wanted to catch bugs? Now you can. Curious about ironworking? It’s finally a thing. There’s even a deep, mysterious cave system to explore, hidden behind ancient sigils. It’s like unlocking ancient cheat codes for yourapeople. The flora and fauna are getting a glow-up too. In the sunny regions, you’ll find things like rye, lentils, and soaproot (yep, you can finally make soap!). Meanwhile, the moonlit lands bring in some adorable new animals, including—you’re not ready—dodos and pandas in 1.3 update. Yes. Freakin’ pandas. You’ll also meet new characters (romance options included, hello Azkel and Sayra 👀), trade for awesome new outfits, build with fresh decorations, and even hit the water on a raft to fish like a prehistoric pro. And if you're wondering about platforms—it’s out everywhere. Steam, Switch, PlayStation, Xbox. Plus, full crossplay co-op support is live, so no matter where your people are gaming, you can play together. Best part? The 1.3 update is totally free. And the base game is 50% off for a week, so it’s the perfect time to hop in if you haven't already. So gather your clan, spark up some new Ideas, and get ready to live your best Pachan life. This Sun and Moon update doesn’t just add content—it evolves the game. Let’s go full Stone Age in this farming sim via Steam and Humble Store. Priced at $12.49 USD / £10.99 / 12,49€ with the 50% discount on Linux, Steam Deck, Mac, and Windows PC.