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Abhimanyu Bhargava

Abhimanyu Bhargava

3 years ago

VeeFriends Series 2: The Biggest NFT Opportunity Ever

More on NFTs & Art

Anton Franzen

Anton Franzen

3 years ago

This is the driving force for my use of NFTs, which will completely transform the world.

Its not a fuc*ing fad.

Photo by kyung on unsplash

It's not about boring monkeys or photos as nfts; that's just what's been pushed up and made a lot of money. The technology underlying those ridiculous nft photos will one day prove your house and automobile ownership and tell you where your banana came from. Are you ready for web3? Soar!

People don't realize that absolutely anything can and will be part of the blockchain and smart contracts, making them even better. I'll tell you a secret: it will and is happening.

Why?

Why is something blockchain-based a good idea? So let’s speak about cars!

So a new Tesla car is manufactured, and when you buy it, it is bound to an NFT on the blockchain that proves current ownership. The NFT in the smart contract can contain some data about the current owner of the car and some data about the car's status, such as the number of miles driven, the car's overall quality, and so on, as well as a reference to a digital document bound to the NFT that has more information.

Now, 40 years from now, if you want to buy a used automobile, you can scan the car's serial number to view its NFT and see all of its history, each owner, how long they owned it, if it had damages, and more. Since it's on the blockchain, it can't be tampered with.

When you're ready to buy it, the owner posts it for sale, you buy it, and it's sent to your wallet. 5 seconds to change owner, 100% safe and verifiable.

Incorporate insurance logic into the car contract. If you crashed, your car's smart contract would take money from your insurance contract and deposit it in an insurance company wallet.

It's limitless. Your funds may be used by investors to provide insurance as they profit from everyone's investments.

Or suppose all car owners in a country deposit a fixed amount of money into an insurance smart contract that promises if something happens, we'll take care of it. It could be as little as $100-$500 per year, and in a country with 10 million people, maybe 3 million would do that, which would be $500 000 000 in that smart contract and it would be used by the insurance company to invest in assets or take a cut, literally endless possibilities.

Instead of $300 per month, you may pay $300 per year to be covered if something goes wrong, and that may include multiple insurances.

What about your grocery store banana, though?

Yes that too.

You can scan a banana to learn its complete history. You'll be able to see where it was cultivated, every middleman in the supply chain, and hopefully the banana's quality, farm, and ingredients used.

If you want locally decent bananas, you can only buy them, offering you transparency and options. I believe it will be an online marketplace where farmers publish their farms and products for trust and transparency. You might also buy bananas from the farmer.

And? Food security to finish the article. If an order of bananas included a toxin, you could easily track down every banana from the same origin and supply chain and uncover the root cause. This is a tremendous thing that will save lives and have a big impact; did you realize that 1 in 6 Americans gets poisoned by food every year? This could lower the number.

To summarize:

Smart contracts can issue nfts as proof of ownership and include functionality.

nft now

nft now

3 years ago

A Guide to VeeFriends and Series 2

VeeFriends is one of the most popular and unique NFT collections. VeeFriends launched around the same time as other PFP NFTs like Bored Ape Yacht Club.

Vaynerchuk (GaryVee) took a unique approach to his large-scale project, which has influenced the NFT ecosystem. GaryVee's VeeFriends is one of the most successful NFT membership use-cases, allowing him to build a community around his creative and business passions.

What is VeeFriends?

GaryVee's NFT collection, VeeFriends, was released on May 11, 2021. VeeFriends [Mini Drops], Book Games, and a forthcoming large-scale "Series 2" collection all stem from the initial drop of 10,255 tokens.

In "Series 1," there are G.O.O. tokens (Gary Originally Owned). GaryVee reserved 1,242 NFTs (over 12% of the supply) for his own collection, so only 9,013 were available at the Series 1 launch.

Each Series 1 token represents one of 268 human traits hand-drawn by Vaynerchuk. Gary Vee's NFTs offer owners incentives.

Who made VeeFriends?

Gary Vaynerchuk, AKA GaryVee, is influential in NFT. Vaynerchuk is the chairman of New York-based communications company VaynerX. Gary Vee, CEO of VaynerMedia, VaynerSports, and bestselling author, is worth $200 million.

GaryVee went from NFT collector to creator, launching VaynerNFT to help celebrities and brands.

Vaynerchuk's influence spans the NFT ecosystem as one of its most prolific voices. He's one of the most influential NFT figures, and his VeeFriends ecosystem keeps growing.

Vaynerchuk, a trend expert, thinks NFTs will be around for the rest of his life and VeeFriends will be a landmark project.

Why use VeeFriends NFTs?

The first VeeFriends collection has sold nearly $160 million via OpenSea. GaryVee insisted that the first 10,255 VeeFriends were just the beginning.

Book Games were announced to the VeeFriends community in August 2021. Mini Drops joined VeeFriends two months later.

Book Games

GaryVee's book "Twelve and a Half: Leveraging the Emotional Ingredients for Business Success" inspired Book Games. Even prior to the announcement Vaynerchuk had mapped out the utility of the book on an NFT scale. Book Games tied his book to the VeeFriends ecosystem and solidified its place in the collection.

GaryVee says Book Games is a layer 2 NFT project with 125,000 burnable tokens. Vaynerchuk's NFT fans were incentivized to buy as many copies of his new book as possible to receive NFT rewards later.

First, a bit about “layer 2.”

Layer 2 blockchain solutions help scale applications by routing transactions away from Ethereum Mainnet (layer 1). These solutions benefit from Mainnet's decentralized security model but increase transaction speed and reduce gas fees.

Polygon (integrated into OpenSea) and Immutable X are popular Ethereum layer 2 solutions. GaryVee chose Immutable X to reduce gas costs (transaction fees). Given the large supply of Book Games tokens, this decision will likely benefit the VeeFriends community, especially if the games run forever.

What's the strategy?

The VeeFriends patriarch announced on Aug. 27, 2021, that for every 12 books ordered during the Book Games promotion, customers would receive one NFT via airdrop. After nearly 100 days, GV sold over a million copies and announced that Book Games would go gamified on Jan. 10, 2022.

Immutable X's trading options make Book Games a "game." Book Games players can trade NFTs for other NFTs, sports cards, VeeCon tickets, and other prizes. Book Games can also whitelist other VeeFirends projects, which we'll cover in Series 2.

VeeFriends Mini Drops

GaryVee launched VeeFriends Mini Drops two months after Book Games, focusing on collaboration, scarcity, and the characters' "cultural longevity."

Spooky Vees, a collection of 31 1/1 Halloween-themed VeeFriends, was released on Halloween. First-come, first-served VeeFriend owners could claim these NFTs.

Mini Drops includes Gift Goat NFTs. By holding the Gift Goat VeeFriends character, collectors will receive 18 exclusive gifts curated by GaryVee and the team. Each gifting experience includes one physical gift and one NFT out of 555, to match the 555 Gift Goat tokens.

Gift Goat holders have gotten NFTs from Danny Cole (Creature World), Isaac "Drift" Wright (Where My Vans Go), Pop Wonder, and more.

GaryVee is poised to release the largest expansion of the VeeFriends and VaynerNFT ecosystem to date with VeeFriends Series 2.

VeeCon 101

By owning VeeFriends NFTs, collectors can join the VeeFriends community and attend VeeCon in 2022. The conference is only open to VeeCon NFT ticket holders (VeeFreinds + possibly more TBA) and will feature Beeple, Steve Aoki, and even Snoop Dogg.

The VeeFreinds floor in 2022 Q1 has remained at 16 ETH ($52,000), making VeeCon unattainable for most NFT enthusiasts. Why would someone spend that much crypto on a Minneapolis "superconference" ticket? Because of Gary Vaynerchuk.

Everything to know about VeeFriends Series 2

Vaynerchuk revealed in April 2022 that the VeeFriends ecosystem will grow by 55,555 NFTs after months of teasing.

With VeeFriends Series 2, each token will cost $995 USD in ETH, allowing NFT enthusiasts to join at a lower cost. The new series will be released on multiple dates in April.

Book Games NFT holders on the Friends List (whitelist) can mint Series 2 NFTs on April 12. Book Games holders have 32,000 NFTs.

VeeFriends Series 1 NFT holders can claim Series 2 NFTs on April 12. This allotment's supply is 10,255, like Series 1's.

On April 25, the public can buy 10,000 Series 2 NFTs. Unminted Friends List NFTs will be sold on this date, so this number may change.

The VeeFriends ecosystem will add 15 new characters (220 tokens each) on April 27. One character will be released per day for 15 days, and the only way to get one is to enter a daily raffle with Book Games tokens.

Series 2 NFTs won't give owners VeeCon access, but they will offer other benefits within the VaynerNFT ecosystem. Book Games and Series 2 will get new token burn mechanics in the upcoming drop.

Visit the VeeFriends blog for the latest collection info.

Where can you buy Gary Vee’s NFTs?

Need a VeeFriend NFT? Gary Vee recommends doing "50 hours of homework" before buying. OpenSea sells VeeFriends NFTs.

Scott Duke Kominers

3 years ago

NFT Creators Go Creative Commons Zero (cc0)


On January 1, "Public Domain Day," thousands of creative works immediately join the public domain. The original creator or copyright holder loses exclusive rights to reproduce, adapt, or publish the work, and anybody can use it. It happens with movies, poems, music, artworks, books (where creative rights endure 70 years beyond the author's death), and sometimes source code.

Public domain creative works open the door to new uses. 400,000 sound recordings from before 1923, including Winnie-the-Pooh, were released this year.  With most of A.A. Milne's 1926 Winnie-the-Pooh characters now available, we're seeing innovative interpretations Milne likely never planned. The ancient hyphenated version of the honey-loving bear is being adapted for a horror movie: "Winnie-the-Pooh: Blood and Honey"... with Pooh and Piglet as the baddies.

Counterintuitively, experimenting and recombination can occasionally increase IP value. Open source movements allow the public to build on (or fork and duplicate) existing technologies. Permissionless innovation helps Android, Linux, and other open source software projects compete. Crypto's success at attracting public development is also due to its support of open source and "remix culture," notably in NFT forums.

Production memes

NFT projects use several IP strategies to establish brands, communities, and content. Some preserve regular IP protections; others offer NFT owners the opportunity to innovate on connected IP; yet others have removed copyright and other IP safeguards.

By using the "Creative Commons Zero" (cc0) license, artists can intentionally select for "no rights reserved." This option permits anyone to benefit from derivative works without legal repercussions. There's still a lot of confusion between copyrights and NFTs, so nothing here should be considered legal, financial, tax, or investment advice. Check out this post for an overview of copyright vulnerabilities with NFTs and how authors can protect owners' rights. This article focuses on cc0.

Nouns, a 2021 project, popularized cc0 for NFTs. Others followed, including: A Common Place, Anonymice, Blitmap, Chain Runners, Cryptoadz, CryptoTeddies, Goblintown, Gradis, Loot, mfers, Mirakai, Shields, and Terrarium Club are cc0 projects.

Popular crypto artist XCOPY licensed their 1-of-1 NFT artwork "Right-click and Save As Guy" under cc0 in January, exactly one month after selling it. cc0 has spawned many derivatives.

"Right-click Save As Guy" by XCOPY (1)/derivative works (2)

"Right-click Save As Guy" by XCOPY (1)/derivative works (2)

XCOPY said Monday he would apply cc0 to "all his existing art." "We haven't seen a cc0 summer yet, but I think it's approaching," said the artist. - predicting a "DeFi summer" in 2020, when decentralized finance gained popularity.

Why do so many NFT authors choose "no rights"?

Promoting expansions of the original project to create a more lively and active community is one rationale. This makes sense in crypto, where many value open sharing and establishing community.

Creativity depends on cultural significance. NFTs may allow verifiable ownership of any digital asset, regardless of license, but cc0 jumpstarts "meme-ability" by actively, not passively, inviting derivative works. As new derivatives are made and shared, attention might flow back to the original, boosting its reputation. This may inspire new interpretations, leading in a flywheel effect where each derivative adds to the original's worth - similar to platform network effects, where platforms become more valuable as more users join them.

cc0 licence allows creators "seize production memes."

"SEASON 1 MEME CARD 2"

Physical items are also using cc0 NFT assets, thus it's not just a digital phenomenon. The Nouns Vision initiative turned the square-framed spectacles shown on each new NounsDAO NFT ("one per day, forever") into luxury sunglasses. Blitmap's pixel-art has been used on shoes, apparel, and caps. In traditional IP regimes, a single owner controls creation, licensing, and production.

The physical "blitcap" (3rd level) is a descendant of the trait in the cc0 Chain Runners collection (2nd), which uses the "logo" from cc0 Blitmap (1st)! The Logo is Blitmap token #84 and has been used as a trait in various collections. The "Dom Rose" is another popular token. These homages reference Blitmap's influence as a cc0 leader, as one of the earliest NFT projects to proclaim public domain intents. A new collection, Citizens of Tajigen, emerged last week with a Blitcap characteristic.

These derivatives can be a win-win for everyone, not just the original inventors, especially when using NFT assets to establish unique brands. As people learn about the derivative, they may become interested in the original. If you see someone wearing Nouns glasses on the street (or in a Super Bowl ad), you may desire a pair, but you may also be interested in buying an original NounsDAO NFT or related derivative.

Blitmap Logo Hat (1), Chain Runners #780 ft. Hat (2), and Blitmap Original "Logo #87" (3)

Blitmap Logo Hat (1), Chain Runners #780 ft. Hat (2), and Blitmap Original "Logo #87" (3)

Co-creating open source

NFTs' power comes from smart contract technology's intrinsic composability. Many smart contracts can be integrated or stacked to generate richer applications.

"Money Legos" describes how decentralized finance ("DeFi") smart contracts interconnect to generate new financial use cases. Yearn communicates with MakerDAO's stablecoin $DAI and exchange liquidity provider Curve by calling public smart contract methods. NFTs and their underlying smart contracts can operate as the base-layer framework for recombining and interconnecting culture and creativity.

cc0 gives an NFT's enthusiast community authority to develop new value layers whenever, wherever, and however they wish.

Multiple cc0 projects are playable characters in HyperLoot, a Loot Project knockoff.

Open source and Linux's rise are parallels. When the internet was young, Microsoft dominated the OS market with Windows. Linux (and its developer Linus Torvalds) championed a community-first mentality, freely available the source code without restrictions. This led to developers worldwide producing new software for Linux, from web servers to databases. As people (and organizations) created world-class open source software, Linux's value proposition grew, leading to explosive development and industry innovation. According to Truelist, Linux powers 96.3% of the top 1 million web servers and 85% of smartphones.

With cc0 licensing empowering NFT community builders, one might hope for long-term innovation. Combining cc0 with NFTs "turns an antagonistic game into a co-operative one," says NounsDAO cofounder punk4156. It's important on several levels. First, decentralized systems from open source to crypto are about trust and coordination, therefore facilitating cooperation is crucial. Second, the dynamics of this cooperation work well in the context of NFTs because giving people ownership over their digital assets allows them to internalize the results of co-creation through the value that accrues to their assets and contributions, which incentivizes them to participate in co-creation in the first place.

Licensed to create

If cc0 projects are open source "applications" or "platforms," then NFT artwork, metadata, and smart contracts provide the "user interface" and the underlying blockchain (e.g., Ethereum) is the "operating system." For these apps to attain Linux-like potential, more infrastructure services must be established and made available so people may take advantage of cc0's remixing capabilities.

These services are developing. Zora protocol and OpenSea's open source Seaport protocol enable open, permissionless NFT marketplaces. A pixel-art-rendering engine was just published on-chain to the Ethereum blockchain and integrated into OKPC and ICE64. Each application improves blockchain's "out-of-the-box" capabilities, leading to new apps created from the improved building blocks.

Web3 developer growth is at an all-time high, yet it's still a small fraction of active software developers globally. As additional developers enter the field, prospective NFT projects may find more creative and infrastructure Legos for cc0 and beyond.

Electric Capital Developer Report (2021), p. 122

Electric Capital Developer Report (2021), p. 122

Growth requires composability. Users can easily integrate digital assets developed on public standards and compatible infrastructure into other platforms. The Loot Project is one of the first to illustrate decentralized co-creation, worldbuilding, and more in NFTs. This example was low-fi or "incomplete" aesthetically, providing room for imagination and community co-creation.

Loot began with a series of Loot bag NFTs, each listing eight "adventure things" in white writing on a black backdrop (such as Loot Bag #5726's "Katana, Divine Robe, Great Helm, Wool Sash, Divine Slippers, Chain Gloves, Amulet, Gold Ring"). Dom Hofmann's free Loot bags served as a foundation for the community.

Several projects have begun metaphorical (lore) and practical (game development) world-building in a short time, with artists contributing many variations to the collective "Lootverse." They've produced games (Realms & The Crypt), characters (Genesis Project, Hyperloot, Loot Explorers), storytelling initiatives (Banners, OpenQuill), and even infrastructure (The Rift).

Why cc0 and composability? Because consumers own and control Loot bags, they may use them wherever they choose by connecting their crypto wallets. This allows users to participate in multiple derivative projects, such as  Genesis Adventurers, whose characters appear in many others — creating a decentralized franchise not owned by any one corporation.

Genesis Project's Genesis Adventurer (1) with HyperLoot (2) and Loot Explorer (3) versions

Genesis Project's Genesis Adventurer (1) with HyperLoot (2) and Loot Explorer (3) versions

When to go cc0

There are several IP development strategies NFT projects can use. When it comes to cc0, it’s important to be realistic. The public domain won't make a project a runaway success just by implementing the license. cc0 works well for NFT initiatives that can develop a rich, enlarged ecosystem.

Many of the most successful cc0 projects have introduced flexible intellectual property. The Nouns brand is as obvious for a beer ad as for real glasses; Loot bags are simple primitives that make sense in all adventure settings; and the Goblintown visual style looks good on dwarfs, zombies, and cranky owls as it does on Val Kilmer.

The ideal cc0 NFT project gives builders the opportunity to add value:

  • vertically, by stacking new content and features directly on top of the original cc0 assets (for instance, as with games built on the Loot ecosystem, among others), and

  • horizontally, by introducing distinct but related intellectual property that helps propagate the original cc0 project’s brand (as with various Goblintown derivatives, among others).

These actions can assist cc0 NFT business models. Because cc0 NFT projects receive royalties from secondary sales, third-party extensions and derivatives can boost demand for the original assets.

Using cc0 license lowers friction that could hinder brand-reinforcing extensions or lead to them bypassing the original. Robbie Broome recently argued (in the context of his cc0 project A Common Place) that giving away his IP to cc0 avoids bad rehashes down the line. If UrbanOutfitters wanted to put my design on a tee, they could use the actual work instead of hiring a designer. CC0 can turn competition into cooperation.

Community agreement about core assets' value and contribution can help cc0 projects. Cohesion and engagement are key. Using the above examples: Developers can design adventure games around whatever themes and item concepts they desire, but many choose Loot bags because of the Lootverse's community togetherness. Flipmap shared half of its money with the original Blitmap artists in acknowledgment of that project's core role in the community. This can build a healthy culture within a cc0 project ecosystem. Commentator NiftyPins said it was smart to acknowledge the people that constructed their universe. Many OG Blitmap artists have popped into the Flipmap discord to share information.

cc0 isn't a one-size-fits-all answer; NFTs formed around well-established brands may prefer more restrictive licenses to preserve their intellectual property and reinforce exclusivity. cc0 has some superficial similarities to permitting NFT owners to market the IP connected with their NFTs (à la Bored Ape Yacht Club), but there is a significant difference: cc0 holders can't exclude others from utilizing the same IP. This can make it tougher for holders to develop commercial brands on cc0 assets or offer specific rights to partners. Holders can still introduce enlarged intellectual property (such as backstories or derivatives) that they control.


Blockchain technologies and the crypto ethos are decentralized and open-source. This makes it logical for crypto initiatives to build around cc0 content models, which build on the work of the Creative Commons foundation and numerous open source pioneers.

NFT creators that choose cc0 must select how involved they want to be in building the ecosystem. Some cc0 project leaders, like Chain Runners' developers, have kept building on top of the initial cc0 assets, creating an environment derivative projects can plug into. Dom Hofmann stood back from Loot, letting the community lead. (Dom is also working on additional cc0 NFT projects for the company he formed to build Blitmap.) Other authors have chosen out totally, like sartoshi, who announced his exit from the cc0 project he founded, mfers, and from the NFT area by publishing a final edition suitably named "end of sartoshi" and then deactivating his Twitter account. A multi-signature wallet of seven mfers controls the project's smart contract. 

cc0 licensing allows a robust community to co-create in ways that benefit all members, regardless of original creators' continuous commitment. We foresee more organized infrastructure and design patterns as NFT matures. Like open source software, value capture frameworks may see innovation. (We could imagine a variant of the "Sleepycat license," which requires commercial software to pay licensing fees when embedding open source components.) As creators progress the space, we expect them to build unique rights and licensing strategies. cc0 allows NFT producers to bootstrap ideas that may take off.

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Florian Wahl

Florian Wahl

3 years ago

An Approach to Product Strategy

I've been pondering product strategy and how to articulate it. Frameworks helped guide our thinking.

If your teams aren't working together or there's no clear path to victory, your product strategy may not be well-articulated or communicated (if you have one).

Before diving into a product strategy's details, it's important to understand its role in the bigger picture — the pieces that move your organization forward.

the overall picture

A product strategy is crucial, in my opinion. It's part of a successful product or business. It's the showpiece.

The Big Picture: Vision, Product Strategy, Goals, Roadmap

To simplify, we'll discuss four main components:

  1. Vision

  2. Product Management

  3. Goals

  4. Roadmap

Vision

Your company's mission? Your company/product in 35 years? Which headlines?

The vision defines everything your organization will do in the long term. It shows how your company impacted the world. It's your organization's rallying cry.

An ambitious but realistic vision is needed.

Without a clear vision, your product strategy may be inconsistent.

Product Management

Our main subject. Product strategy connects everything. It fulfills the vision.

In Part 2, we'll discuss product strategy.

Goals

This component can be goals, objectives, key results, targets, milestones, or whatever goal-tracking framework works best for your organization.

These product strategy metrics will help your team prioritize strategies and roadmaps.

Your company's goals should be unified. This fuels success.

Roadmap

The roadmap is your product strategy's timeline. It provides a prioritized view of your team's upcoming deliverables.

A roadmap is time-bound and includes measurable goals for your company. Your team's steps and capabilities for executing product strategy.

If your team has trouble prioritizing or defining a roadmap, your product strategy or vision is likely unclear.

Formulation of a Product Strategy

Now that we've discussed where your product strategy fits in the big picture, let's look at a framework.

Product Strategy Framework: Challenges, Decided Approach, Actions

A product strategy should include challenges, an approach, and actions.

Challenges

First, analyze the problems/situations you're solving. It can be customer- or company-focused.

The analysis should explain the problems and why they're important. Try to simplify the situation and identify critical aspects.

Some questions:

  • What issues are we attempting to resolve?

  • What obstacles—internal or otherwise—are we attempting to overcome?

  • What is the opportunity, and why should we pursue it, in your opinion?

Decided Method

Second, describe your approach. This can be a set of company policies for handling the challenge. It's the overall approach to the first part's analysis.

The approach can be your company's bets, the solutions you've found, or how you'll solve the problems you've identified.

Again, these questions can help:

  • What is the value that we hope to offer to our clients?

  • Which market are we focusing on first?

  • What makes us stand out? Our benefit over rivals?

Actions

Third, identify actions that result from your approach. Second-part actions should be these.

Coordinate these actions. You may need to add products or features to your roadmap, acquire new capabilities through partnerships, or launch new marketing campaigns. Whatever fits your challenges and strategy.

Final questions:

  • What skills do we need to develop or obtain?

  • What is the chosen remedy? What are the main outputs?

  • What else ought to be added to our road map?

Put everything together

… and iterate!

Strategy isn't one-and-done. Changes occur. Economies change. Competitors emerge. Customer expectations change.

One unexpected event can make strategies obsolete quickly. Muscle it. Review, evaluate, and course-correct your strategies with your teams. Quarterly works. In a new or unstable industry, more often.

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

William Anderson

William Anderson

3 years ago

When My Remote Leadership Skills Took Off

4 Ways To Manage Remote Teams & Employees

The wheels hit the ground as I landed in Rochester.

Our six-person satellite office was now part of my team.

Their manager only reported to me the day before, but I had my ticket booked ahead of time.

I had managed remote employees before but this was different. Engineers dialed into headquarters for every meeting.

So when I learned about the org chart change, I knew a strong first impression would set the tone for everything else.

I was either their boss, or their boss's boss, and I needed them to know I was committed.

Managing a fleet of satellite freelancers or multiple offices requires treating others as more than just a face behind a screen.

You must comprehend each remote team member's perspective and daily interactions.

The good news is that you can start using these techniques right now to better understand and elevate virtual team members.

1. Make Visits To Other Offices

If budgeted, visit and work from offices where teams and employees report to you. Only by living alongside them can one truly comprehend their problems with communication and other aspects of modern life.

2. Have Others Come to You

• Having remote, distributed, or satellite employees and teams visit headquarters every quarter or semi-quarterly allows the main office culture to rub off on them.

When remote team members visit, more people get to meet them, which builds empathy.

If you can't afford to fly everyone, at least bring remote managers or leaders. Hopefully they can resurrect some culture.

3. Weekly Work From Home

No home office policy?

Make one.

WFH is a team-building, problem-solving, and office-viewing opportunity.

For dial-in meetings, I started working from home on occasion.

It also taught me which teams “forget” or “skip” calls.

As a remote team member, you experience all the issues first hand.

This isn't as accurate for understanding teams in other offices, but it can be done at any time.

4. Increase Contact Even If It’s Just To Chat

Don't underestimate office banter.

Sometimes it's about bonding and trust, other times it's about business.

If you get all this information in real-time, please forward it.

Even if nothing critical is happening, call remote team members to check in and chat.

I guarantee that building relationships and rapport will increase both their job satisfaction and yours.