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Jennifer Tieu

Jennifer Tieu

3 years ago

Why I Love Azuki


Azuki Banner (www.azuki.com)

Disclaimer: This is my personal viewpoint. I'm not on the Azuki team. Please keep in mind that I am merely a fan, community member, and holder. Please do your own research and pardon my grammar. Thanks!

Azuki has changed my view of NFTs.

When I first entered the NFT world, I had no idea what to expect. I liked the idea. So I invested in some projects, fought for whitelists, and discovered some cool NFTs projects (shout-out to CATC). I lost more money than I earned at one point, but I hadn't invested excessively (only put in what you can afford to lose). Despite my losses, I kept looking. I almost waited for the “ah-ha” moment. A NFT project that changed my perspective on NFTs. What makes an NFT project more than a work of art?

Answer: Azuki.

The Art

The Azuki art drew me in as an anime fan. It looked like something out of an anime, and I'd never seen it before in NFT.
The project was still new. The first two animated teasers were released with little fanfare, but I was impressed with their quality. You can find them on Instagram or in their earlier Tweets.

The teasers hinted that this project could be big and that the team could deliver. It was amazing to see Shao cut the Azuki posters with her katana. Especially at the end when she sheaths her sword and the music cues. Then the live action video of the young boy arranging the Azuki posters seemed movie-like. I felt like I was entering the Azuki story, brand, and dope theme.

The team did not disappoint with the Azuki NFTs. The level of detail in the art is stunning. There were Azukis of all genders, skin and hair types, and more. These 10,000 Azukis have so much representation that almost anyone can find something that resonates. Rather than me rambling on, I suggest you visit the Azuki gallery

The Team

If the art is meant to draw you in and be the project's face, the team makes it more. The NFT would be a JPEG without a good team leader. Not that community isn't important, but no community would rally around a bad team.

Because I've been rugged before, I'm very focused on the team when considering a project. While many project teams are anonymous, I try to find ones that are doxxed (public) or at least appear to be established. Unlike Azuki, where most of the Azuki team is anonymous, Steamboy is public. He is (or was) Overwatch's character art director and co-creator of Azuki. I felt reassured and could trust the project after seeing someone from a major game series on the team.

Then I tried to learn as much as I could about the team. Following everyone on Twitter, reading their tweets, and listening to recorded AMAs. I was impressed by the team's professionalism and dedication to their vision for Azuki, led by ZZZAGABOND.
I believe the phrase “actions speak louder than words” applies to Azuki. I can think of a few examples of what the Azuki team has done, but my favorite is ERC721A.

With ERC721A, Azuki has created a new algorithm that allows minting multiple NFTs for essentially the same cost as minting one NFT.

I was ecstatic when the dev team announced it. This fascinates me as a self-taught developer. Azuki released a product that saves people money, improves the NFT space, and is open source. It showed their love for Azuki and the NFT community.

The Community

Community, community, community. It's almost a chant in the NFT space now. A community, like a team, can make or break a project. We are the project's consumers, shareholders, core, and lifeblood. The team builds the house, and we fill it. We stay for the community.
When I first entered the Azuki Discord, I was surprised by the calm atmosphere. There was no news about the project. No release date, no whitelisting requirements. No grinding or spamming either. People just wanted to hangout, get to know each other, and talk. It was nice. So the team could pick genuine people for their mintlist (aka whitelist).
But nothing fundamental has changed since the release. It has remained an authentic, fun, and helpful community. I'm constantly logging into Discord to chat with others or follow conversations. I see the community's openness to newcomers. Everyone respects each other (barring a few bad apples) and the variety of people passing through is fascinating. This human connection and interaction is what I enjoy about this place. Being a part of a group that supports a cause.
Finally, I want to thank the amazing Azuki mod team and the kissaten channel for their contributions.

The Brand

So, what sets Azuki apart from other projects? They are shaping a brand or identity. The Azuki website, I believe, best captures their vision. (This is me gushing over the site.)

If you go to the website, turn on the dope playlist in the bottom left. The playlist features a mix of Asian and non-Asian hip-hop and rap artists, with some lo-fi thrown in. The songs on the playlist change, but I think you get the vibe Azuki embodies just by turning on the music.
The Garden is our next stop where we are introduced to Azuki.

A brand.

We're creating a new brand together.
A metaverse brand. By the people.
A collection of 10,000 avatars that grant Garden membership. It starts with exclusive streetwear collabs, NFT drops, live events, and more. Azuki allows for a new media genre that the world has yet to discover. Let's build together an Azuki, your metaverse identity.
The Garden is a magical internet corner where art, community, and culture collide. The boundaries between the physical and digital worlds are blurring.
Try a Red Bean.

The text begins with Azuki's intention in the space. It's a community-made metaverse brand. Then it goes into more detail about Azuki's plans. Initiation of a story or journey. "Would you like to take the red bean and jump down the rabbit hole with us?" I love the Matrix red pill or blue pill play they used. (Azuki in Japanese means red bean.)

Morpheus, the rebel leader, offers Neo the choice of a red or blue pill in The Matrix. “You take the blue pill... After the story, you go back to bed and believe whatever you want. Your red pill... Let me show you how deep the rabbit hole goes.” Aware that the red pill will free him from the enslaving control of the machine-generated dream world and allow him to escape into the real world, he takes it. However, living the “truth of reality” is harsher and more difficult.

It's intriguing and draws you in. Taking the red bean causes what? Where am I going? I think they did well in piqueing a newcomer's interest.
Not convinced by the Garden? Read the Manifesto. It reinforces Azuki's role.

Here comes a new wave…
And surfing here is different.
Breaking down barriers.
Building open communities.
Creating magic internet money with our friends.
To those who don’t get it, we tell them: gm.
They’ll come around eventually.
Here’s to the ones with the courage to jump down a peculiar rabbit hole.
One that pulls you away from a world that’s created by many and owned by few…
To a world that’s created by more and owned by all.
From The Garden come the human beans that sprout into your family.
We rise together.
We build together.
We grow together.
Ready to take the red bean?

Not to mention the Mindmap, it sets Azuki apart from other projects and overused Roadmaps. I like how the team recognizes that the NFT space is not linear. So many of us are still trying to figure it out. It is Azuki's vision to adapt to changing environments while maintaining their values. I admire their commitment to long-term growth.

Conclusion

To be honest, I have no idea what the future holds. Azuki is still new and could fail. But I'm a long-term Azuki fan. I don't care about quick gains. The future looks bright for Azuki. I believe in the team's output. I love being an Azuki.
Thank you! IKUZO!

Full post here

Sam Hickmann

Sam Hickmann

3 years ago

+1 💪

More on NFTs & Art

Vishal Chawla

Vishal Chawla

3 years ago

5 Bored Apes borrowed to claim $1.1 million in APE tokens

Takeaway
Unknown user took advantage of the ApeCoin airdrop to earn $1.1 million.
He used a flash loan to borrow five BAYC NFTs, claim the airdrop, and repay the NFTs.

Yuga Labs, the creators of BAYC, airdropped ApeCoin (APE) to anyone who owns one of their NFTs yesterday.

For the Bored Ape Yacht Club and Mutant Ape Yacht Club collections, the team allocated 150 million tokens, or 15% of the total ApeCoin supply, worth over $800 million. Each BAYC holder received 10,094 tokens worth $80,000 to $200,000.

But someone managed to claim the airdrop using NFTs they didn't own. They used the airdrop's specific features to carry it out. And it worked, earning them $1.1 million in ApeCoin.

The trick was that the ApeCoin airdrop wasn't based on who owned which Bored Ape at a given time. Instead, anyone with a Bored Ape at the time of the airdrop could claim it. So if you gave someone your Bored Ape and you hadn't claimed your tokens, they could claim them.

The person only needed to get hold of some Bored Apes that hadn't had their tokens claimed to claim the airdrop. They could be returned immediately.

So, what happened?

The person found a vault with five Bored Ape NFTs that hadn't been used to claim the airdrop.

A vault tokenizes an NFT or a group of NFTs. You put a bunch of NFTs in a vault and make a token. This token can then be staked for rewards or sold (representing part of the value of the collection of NFTs). Anyone with enough tokens can exchange them for NFTs.

This vault uses the NFTX protocol. In total, it contained five Bored Apes: #7594, #8214, #9915, #8167, and #4755. Nobody had claimed the airdrop because the NFTs were locked up in the vault and not controlled by anyone.

The person wanted to unlock the NFTs to claim the airdrop but didn't want to buy them outright s o they used a flash loan, a common tool for large DeFi hacks. Flash loans are a low-cost way to borrow large amounts of crypto that are repaid in the same transaction and block (meaning that the funds are never at risk of not being repaid).

With a flash loan of under $300,000 they bought a Bored Ape on NFT marketplace OpenSea. A large amount of the vault's token was then purchased, allowing them to redeem the five NFTs. The NFTs were used to claim the airdrop, before being returned, the tokens sold back, and the loan repaid.

During this process, they claimed 60,564 ApeCoin airdrops. They then sold them on Uniswap for 399 ETH ($1.1 million). Then they returned the Bored Ape NFT used as collateral to the same NFTX vault.

Attack or arbitrage?

However, security firm BlockSecTeam disagreed with many social media commentators. A flaw in the airdrop-claiming mechanism was exploited, it said.

According to BlockSecTeam's analysis, the user took advantage of a "vulnerability" in the airdrop.

"We suspect a hack due to a flaw in the airdrop mechanism. The attacker exploited this vulnerability to profit from the airdrop claim" said BlockSecTeam.

For example, the airdrop could have taken into account how long a person owned the NFT before claiming the reward.

Because Yuga Labs didn't take a snapshot, anyone could buy the NFT in real time and claim it. This is probably why BAYC sales exploded so soon after the airdrop announcement.

Stephen Moore

Stephen Moore

3 years ago

Trading Volume on OpenSea Drops by 99% as the NFT Boom Comes to an End

Wasn't that a get-rich-quick scheme?

Bored Ape, edited by author

OpenSea processed $2.7 billion in NFT transactions in May 2021.

Fueled by a crypto bull run, rumors of unfathomable riches, and FOMO, Bored Apes, Crypto Punks, and other JPEG-format trash projects flew off the virtual shelves, snatched up by retail investors and celebrities alike.

Over a year later, those shelves are overflowing and warehouses are backlogged. Since March, I've been writing less. In May and June, the bubble was close to bursting.

Apparently, the boom has finally peaked.

This bubble has punctured, and deflation has begun. On Aug. 28, OpenSea processed $9.34 million.

From that euphoric high of $2.7 billion, $9.34 million represents a spectacular decline of 99%.

OpenSea contradicts the data. A trading platform spokeswoman stated the comparison is unfair because it compares the site's highest and lowest trading days. They're the perfect two data points to assess the drop. OpenSea chooses to use ETH volume measures, which ignore crypto's shifting price. Since January 2022, monthly ETH volume has dropped 140%, according to Dune.

Unconvincing counterargument.

Further OpenSea indicators point to declining NFT demand:

  • Since January 2022, daily user visits have decreased by 50%.

  • Daily transactions have decreased by 50% since the beginning of the year in the same manner.

Off-platform, the floor price of Bored Apes has dropped from 145 ETH to 77 ETH. (At $4,800, a reduction from $700,000 to $370,000). Google search data shows waning popular interest.

Data: Google Trends

It is a trend that will soon vanish, just like laser eyes.

NFTs haven't moved since the new year. Eminem and Snoop Dogg can utilize their apes in music videos or as 3D visuals to perform at the VMAs, but the reality is that NFTs have lost their public appeal and the market is trying to regain its footing.

They've lost popularity because?

Breaking records. The technology still lacks genuine use cases a year and a half after being popular.

They're pricey prestige symbols that have made a few people rich through cunning timing or less-than-savory scams or rug pulling. Over $10.5 billion has been taken through frauds, most of which are NFT enterprises promising to be the next Bored Apes, according to Web3 is going wonderfully. As the market falls, many ordinary investors realize they purchased into a self-fulfilling ecosystem that's halted. Many NFTs are sold between owner-held accounts to boost their price, data suggests. Most projects rely on social media excitement to debut with a high price before the first owners sell and chuckle to the bank. When they don't, the initiative fails, leaving investors high and dry.

NFTs are fading like laser eyes. Most people pushing the technology don't believe in it or the future it may bring. No, they just need a Kool-Aid-drunk buyer.

Everybody wins. When your JPEGs are worth 99% less than when you bought them, you've lost.

When demand reaches zero, many will lose.

Yogita Khatri

Yogita Khatri

3 years ago

Moonbirds NFT sells for $1 million in first week

On Saturday, Moonbird #2642, one of the collection's rarest NFTs, sold for a record 350 ETH (over $1 million) on OpenSea.

The Sandbox, a blockchain-based gaming company based in Hong Kong, bought the piece. The seller, "oscuranft" on OpenSea, made around $600,000 after buying the NFT for 100 ETH a week ago.

Owl avatars

Moonbirds is a 10,000 owl NFT collection. It is one of the quickest collections to achieve bluechip status. Proof, a media startup founded by renowned VC Kevin Rose, launched Moonbirds on April 16.

Rose is currently a partner at True Ventures, a technology-focused VC firm. He was a Google Ventures general partner and has 1.5 million Twitter followers.

Rose has an NFT podcast on Proof. It follows Proof Collective, a group of 1,000 NFT collectors and artists, including Beeple, who hold a Proof Collective NFT and receive special benefits.

These include early access to the Proof podcast and in-person events.

According to the Moonbirds website, they are "the official Proof PFP" (picture for proof).

Moonbirds NFTs sold nearly $360 million in just over a week, according to The Block Research and Dune Analytics. Its top ten sales range from $397,000 to $1 million.

In the current market, Moonbirds are worth 33.3 ETH. Each NFT is 2.5 ETH. Holders have gained over 12 times in just over a week.

Why was it so popular?

The Block Research's NFT analyst, Thomas Bialek, attributes Moonbirds' rapid rise to Rose's backing, the success of his previous Proof Collective project, and collectors' preference for proven NFT projects.

Proof Collective NFT holders have made huge gains. These NFTs were sold in a Dutch auction last December for 5 ETH each. According to OpenSea, the current floor price is 109 ETH.

According to The Block Research, citing Dune Analytics, Proof Collective NFTs have sold over $39 million to date.

Rose has bigger plans for Moonbirds. Moonbirds is introducing "nesting," a non-custodial way for holders to stake NFTs and earn rewards.

Holders of NFTs can earn different levels of status based on how long they keep their NFTs locked up.

"As you achieve different nest status levels, we can offer you different benefits," he said. "We'll have in-person meetups and events, as well as some crazy airdrops planned."

Rose went on to say that Proof is just the start of "a multi-decade journey to build a new media company."

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Jack Burns

Jack Burns

3 years ago

Here's what to expect from NASA Artemis 1 and why it's significant.

NASA's Artemis 1 mission will help return people to the Moon after a half-century break. The mission is a shakedown cruise for NASA's Space Launch System and Orion Crew Capsule.

The spaceship will visit the Moon, deploy satellites, and enter orbit. NASA wants to practice operating the spacecraft, test the conditions people will face on the Moon, and ensure a safe return to Earth.

We asked Jack Burns, a space scientist at the University of Colorado Boulder and former member of NASA's Presidential Transition Team, to describe the mission, explain what the Artemis program promises for space exploration, and reflect on how the space program has changed in the half-century since humans last set foot on the moon.

What distinguishes Artemis 1 from other rockets?

Artemis 1 is the Space Launch System's first launch. NASA calls this a "heavy-lift" vehicle. It will be more powerful than Apollo's Saturn V, which transported people to the Moon in the 1960s and 1970s.

It's a new sort of rocket system with two strap-on solid rocket boosters from the space shuttle. It's a mix of the shuttle and Saturn V.

The Orion Crew Capsule will be tested extensively. It'll spend a month in the high-radiation Moon environment. It will also test the heat shield, which protects the capsule and its occupants at 25,000 mph. The heat shield must work well because this is the fastest capsule descent since Apollo.

This mission will also carry miniature Moon-orbiting satellites. These will undertake vital precursor science, including as examining further into permanently shadowed craters where scientists suspect there is water and measuring the radiation environment to see long-term human consequences.

Diagram depicting earth, moon, and spacecraft travel route

Artemis 1 will launch, fly to the Moon, place satellites, orbit it, return to Earth, and splash down in the ocean. NASA.

What's Artemis's goal? What launches are next?

The mission is a first step toward Artemis 3, which will lead to the first human Moon missions since 1972. Artemis 1 is unmanned.

Artemis 2 will have astronauts a few years later. Like Apollo 8, it will be an orbital mission that circles the Moon and returns. The astronauts will orbit the Moon longer and test everything with a crew.

Eventually, Artemis 3 will meet with the SpaceX Starship on the Moon's surface and transfer people. Orion will stay in orbit while the lunar Starship lands astronauts. They'll go to the Moon's south pole to investigate the water ice there.

Artemis is reminiscent of Apollo. What's changed in 50 years?

Kennedy wanted to beat the Soviets to the Moon with Apollo. The administration didn't care much about space flight or the Moon, but the goal would place America first in space and technology.

You live and die by the sword if you do that. When the U.S. reached the Moon, it was over. Russia lost. We planted flags and did science experiments. Richard Nixon canceled the program after Apollo 11 because the political goals were attained.

Large rocket with two boosters between two gates

NASA's new Space Launch System is brought to a launchpad. NASA

50 years later... It's quite different. We're not trying to beat the Russians, Chinese, or anyone else, but to begin sustainable space exploration.

Artemis has many goals. It includes harnessing in-situ resources like water ice and lunar soil to make food, fuel, and building materials.

SpaceX is part of this first journey to the Moon's surface, therefore the initiative is also helping to develop a lunar and space economy. NASA doesn't own the Starship but is buying seats for astronauts. SpaceX will employ Starship to transport cargo, private astronauts, and foreign astronauts.

Fifty years of technology advancement has made getting to the Moon cheaper and more practical, and computer technology allows for more advanced tests. 50 years of technological progress have changed everything. Anyone with enough money can send a spacecraft to the Moon, but not humans.

Commercial Lunar Payload Services engages commercial companies to develop uncrewed Moon landers. We're sending a radio telescope to the Moon in January. Even 10 years ago, that was impossible.

Since humans last visited the Moon 50 years ago, technology has improved greatly.

What other changes does Artemis have in store?

The government says Artemis 3 will have at least one woman and likely a person of color. 

I'm looking forward to seeing more diversity so young kids can say, "Hey, there's an astronaut that looks like me. I can do this. I can be part of the space program.

Andy Walker

Andy Walker

2 years ago

Why personal ambition and poor leadership caused Google layoffs

Google announced 6% layoffs recently (or 12,000 people). This aligns it with most tech companies. A publicly contrite CEO explained that they had overhired during the COVID-19 pandemic boom and had to address it, but they were sorry and took full responsibility. I thought this was "bullshit" too. Meta, Amazon, Microsoft, and others must feel similarly. I spent 10 years at Google, and these things don't reflect well on the company's leaders.

All publicly listed companies have a fiduciary duty to act in the best interests of their shareholders. Dodge vs. Ford Motor Company established this (1919). Henry Ford wanted to reduce shareholder payments to offer cheaper cars and better wages. Ford stated.

My ambition is to employ still more men, to spread the benefits of this industrial system to the greatest possible number, to help them build up their lives and their homes. To do this we are putting the greatest share of our profits back in the business.

The Dodge brothers, who owned 10% of Ford, opposed this and sued Ford for the payments to start their own company. They won, preventing Ford from raising prices or salaries. If you have a vocal group of shareholders with the resources to sue you, you must prove you are acting in their best interests. Companies prioritize shareholders. Giving activist investors a stick to threaten you almost enshrines short-term profit over long-term thinking.

This underpins Google's current issues. Institutional investors who can sue Google see it as a wasteful company they can exploit. That doesn't mean you have to maximize profits (thanks to those who pointed out my ignorance of US corporate law in the comments and on HN), but it allows pressure. I feel for those navigating this. This is about unrestrained capitalism.

When Google went public, Larry Page and Sergey Brin knew the risks and worked hard to keep control. In their Founders' Letter to investors, they tried to set expectations for the company's operations.

Our long-term focus as a private company has paid off. Public companies do the same. We believe outside pressures lead companies to sacrifice long-term opportunities to meet quarterly market expectations.

The company has transformed since that letter. The company has nearly 200,000 full-time employees and a trillion-dollar market cap. Large investors have bought company stock because it has been a good long-term bet. Why are they restless now?

Other big tech companies emerged and fought for top talent. This has caused rising compensation packages. Google has also grown rapidly (roughly 22,000 people hired to the end of 2022). At $300,000 median compensation, those 22,000 people added $6.6 billion in salary overheads in 2022. Exorbitant. If the company still makes $16 billion every quarter, maybe not. Investors wonder if this value has returned.

Investors are right. Google uses people wastefully. However, by bluntly reducing headcount, they're not addressing the root causes and hurting themselves. No studies show that downsizing this way boosts productivity. There is plenty of evidence that they'll lose out because people will be risk-averse and distrust their leadership.

The company's approach also stinks. Finding out that you no longer have a job because you can’t log in anymore (sometimes in cases where someone is on call for protecting your production systems) is no way to fire anyone. Being with a narcissistic sociopath is like being abused. First, you receive praise and fancy perks for making the cut. You're fired by text and ghosted. You're told to appreciate the generous severance package. This firing will devastate managers and teams. This type of firing will take years to recover self-esteem. Senior management contributed to this. They chose the expedient answer, possibly by convincing themselves they were managing risk and taking the Macbeth approach of “If it were done when ’tis done, then ’twere well It were done quickly”.

Recap. Google's leadership did a stupid thing—mass firing—in a stupid way. How do we get rid of enough people to make investors happier? and "have 6% less people." Empathetic leaders should not emulate Elon Musk. There is no humane way to fire 12,000 people, but there are better ways. Why is Google so wasteful?

Ambition answers this. There aren't enough VP positions for a group of highly motivated, ambitious, and (increasingly) ruthless people. I’ve loitered around the edges of this world and a large part of my value was to insulate my teams from ever having to experience it. It’s like Game of Thrones played out through email and calendar and over video call.

Your company must look a certain way to be promoted to director or higher. You need the right people at the right levels under you. Long-term, growing your people will naturally happen if you're working on important things. This takes time, and you're never more than 6–18 months from a reorg that could start you over. Ambitious people also tend to be impatient. So, what do you do?

Hiring and vanity projects. To shape your company, you hire at the right levels. You value vanity metrics like active users over product utility. Your promo candidates get through by subverting the promotion process. In your quest for growth, you avoid performance managing people out. You avoid confronting toxic peers because you need their support for promotion. Your cargo cult gets you there.

Its ease makes Google wasteful. Since they don't face market forces, the employees don't see it as a business. Why would you do when the ads business is so profitable? Complacency causes senior leaders to prioritize their own interests. Empires collapse. Personal ambition often trumped doing the right thing for users, the business, or employees. Leadership's ambition over business is the root cause. Vanity metrics, mass hiring, and vague promises have promoted people to VP. Google goes above and beyond to protect senior leaders.

The decision-makers and beneficiaries are not the layoffees. Stock price increase beneficiaries. The people who will post on LinkedIn how it is about misjudging the market and how they’re so sorry and take full responsibility. While accumulating wealth, the dark room dwellers decide who stays and who goes. The billionaire investors. Google should start by addressing its bloated senior management, but — as they say — turkeys don't vote for Christmas. It should examine its wastefulness and make tough choices to fix it. A 6% cut is a blunt tool that admits you're not running your business properly. why aren’t the people running the business the ones shortly to be entering the job market?

This won't fix Google's wastefulness. The executives may never regain trust after their approach. Suppressed creativity. Business won't improve. Google will have lost its founding vision and us all. Large investors know they can force Google's CEO to yield. The rich will get richer and rationalize leaving 12,000 people behind. Cycles repeat.

It doesn’t have to be this way. In 2013, Nintendo's CEO said he wouldn't fire anyone for shareholders. Switch debuted in 2017. Nintendo's stock has increased by nearly five times, or 19% a year (including the drop most of the stock market experienced last year). Google wasted 12,000 talented people. To please rich people.

Vitalik

Vitalik

4 years ago

An approximate introduction to how zk-SNARKs are possible (part 2)

If tasked with the problem of coming up with a zk-SNARK protocol, many people would make their way to this point and then get stuck and give up. How can a verifier possibly check every single piece of the computation, without looking at each piece of the computation individually? But it turns out that there is a clever solution.

Polynomials

Polynomials are a special class of algebraic expressions of the form:

  • x+5
  • x^4
  • x^3+3x^2+3x+1
  • 628x^{271}+318x^{270}+530x^{269}+…+69x+381

i.e. they are a sum of any (finite!) number of terms of the form cx^k

There are many things that are fascinating about polynomials. But here we are going to zoom in on a particular one: polynomials are a single mathematical object that can contain an unbounded amount of information (think of them as a list of integers and this is obvious). The fourth example above contained 816 digits of tau, and one can easily imagine a polynomial that contains far more.

Furthermore, a single equation between polynomials can represent an unbounded number of equations between numbers. For example, consider the equation A(x)+ B(x) = C(x). If this equation is true, then it's also true that:

  • A(0)+B(0)=C(0)
  • A(1)+B(1)=C(1)
  • A(2)+B(2)=C(2)
  • A(3)+B(3)=C(3)

And so on for every possible coordinate. You can even construct polynomials to deliberately represent sets of numbers so you can check many equations all at once. For example, suppose that you wanted to check:

  • 12+1=13
  • 10+8=18
  • 15+8=23
  • 15+13=28

You can use a procedure called Lagrange interpolation to construct polynomials A(x) that give (12,10,15,15) as outputs at some specific set of coordinates (eg. (0,1,2,3)), B(x) the outputs (1,8,8,13) on thos same coordinates, and so forth. In fact, here are the polynomials:

  • A(x)=-2x^3+\frac{19}{2}x^2-\frac{19}{2}x+12
  • B(x)=2x^3-\frac{19}{2}x^2+\frac{29}{2}x+1
  • C(x)=5x+13

Checking the equation A(x)+B(x)=C(x) with these polynomials checks all four above equations at the same time.

Comparing a polynomial to itself

You can even check relationships between a large number of adjacent evaluations of the same polynomial using a simple polynomial equation. This is slightly more advanced. Suppose that you want to check that, for a given polynomial F, F(x+2)=F(x)+F(x+1) with the integer range {0,1…89} (so if you also check F(0)=F(1)=1, then F(100) would be the 100th Fibonacci number)

As polynomials, F(x+2)-F(x+1)-F(x) would not be exactly zero, as it could give arbitrary answers outside the range x={0,1…98}. But we can do something clever. In general, there is a rule that if a polynomial P is zero across some set S=\{x_1,x_2…x_n\} then it can be expressed as P(x)=Z(x)*H(x), where Z(x)=(x-x_1)*(x-x_2)*…*(x-x_n) and H(x) is also a polynomial. In other words, any polynomial that equals zero across some set is a (polynomial) multiple of the simplest (lowest-degree) polynomial that equals zero across that same set.

Why is this the case? It is a nice corollary of polynomial long division: the factor theorem. We know that, when dividing P(x) by Z(x), we will get a quotient Q(x) and a remainder R(x) is strictly less than that of Z(x). Since we know that P is zero on all of S, it means that R has to be zero on all of S as well. So we can simply compute R(x) via polynomial interpolation, since it's a polynomial of degree at most n-1 and we know n values (the zeros at S). Interpolating a polynomial with all zeroes gives the zero polynomial, thus R(x)=0 and H(x)=Q(x).

Going back to our example, if we have a polynomial F that encodes Fibonacci numbers (so F(x+2)=F(x)+F(x+1) across x=\{0,1…98\}), then I can convince you that F actually satisfies this condition by proving that the polynomial P(x)=F(x+2)-F(x+1)-F(x) is zero over that range, by giving you the quotient:
H(x)=\frac{F(x+2)-F(x+1)-F(x)}{Z(x)}
Where Z(x) = (x-0)*(x-1)*…*(x-98).
You can calculate Z(x) yourself (ideally you would have it precomputed), check the equation, and if the check passes then F(x) satisfies the condition!

Now, step back and notice what we did here. We converted a 100-step-long computation into a single equation with polynomials. Of course, proving the N'th Fibonacci number is not an especially useful task, especially since Fibonacci numbers have a closed form. But you can use exactly the same basic technique, just with some extra polynomials and some more complicated equations, to encode arbitrary computations with an arbitrarily large number of steps.

see part 3