More on NFTs & Art

Sea Launch
3 years ago
A guide to NFT pre-sales and whitelists
Before we dig through NFT whitelists and pre-sales, if you know absolutely nothing about NFTs, check our NFT Glossary.
What are pre-sales and whitelists on NFTs?
An NFT pre-sale, as the name implies, allows community members or early supporters of an NFT project to mint before the public, usually via a whitelist or mint pass.
Coin collectors can use mint passes to claim NFTs during the public sale. Because the mint pass is executed by “burning” an NFT into a specific crypto wallet, the collector is not concerned about gas price spikes.
A whitelist is used to approve a crypto wallet address for an NFT pre-sale. In a similar way to an early access list, it guarantees a certain number of crypto wallets can mint one (or more) NFT.
New NFT projects can do a pre-sale without a whitelist, but whitelists are good practice to avoid gas wars and a fair shot at minting an NFT before launching in competitive NFT marketplaces like Opensea, Magic Eden, or CNFT.
Should NFT projects do pre-sales or whitelists? 👇
The reasons to do pre-sales or a whitelist for NFT creators:
Time the market and gain traction.
Pre-sale or whitelists can help NFT projects gauge interest early on.
Whitelist spots filling up quickly is usually a sign of a successful launch, though it does not guarantee NFT longevity (more on that later). Also, full whitelists create FOMO and momentum for the public sale among non-whitelisted NFT collectors.
If whitelist signups are low or slow, projects may need to work on their vision, community, or product. Or the market is in a bear cycle. In either case, it aids NFT projects in market timing.
Reward the early NFT Community members.
Pre-sale and whitelists can help NFT creators reward early supporters.
First, by splitting the minting process into two phases, early adopters get a chance to mint one or more NFTs from their collection at a discounted or even free price.
Did you know that BAYC started at 0.08 eth each? A serum that allowed you to mint a Mutant Ape has become as valuable as the original BAYC.
(2) Whitelists encourage early supporters to help build a project's community in exchange for a slot or status. If you invite 10 people to the NFT Discord community, you get a better ranking or even a whitelist spot.
Pre-sale and whitelisting have become popular ways for new projects to grow their communities and secure future buyers.
Prevent gas wars.
Most new NFTs are created on the Ethereum blockchain, which has the highest transaction fees (also known as gas) (Solana, Cardano, Polygon, Binance Smart Chain, etc).
An NFT public sale is a gas war when a large number of NFT collectors (or bots) try to mint an NFT at the same time.
Competing collectors are willing to pay higher gas fees to prioritize their transaction and out-price others when upcoming NFT projects are hyped and very popular.
Pre-sales and whitelisting prevent gas wars by breaking the minting process into smaller batches of members or season launches.
The reasons to do pre-sales or a whitelists for NFT collectors:
How do I get on an NFT whitelist?
- Popular NFT collections act as a launchpad for other new or hyped NFT collections.
Example: Interfaces NFTs gives out 100 whitelist spots to Deadfellaz NFTs holders. Both NFT projects win. Interfaces benefit from Deadfellaz's success and brand equity.
In this case, to get whitelisted NFT collectors need to hold that specific NFT that is acting like a launchpad.
- A NFT studio or collection that launches a new NFT project and rewards previous NFT holders with whitelist spots or pre-sale access.
The whitelist requires previous NFT holders or community members.
NFT Alpha Groups are closed, small, tight-knit Discord servers where members share whitelist spots or giveaways from upcoming NFTs.
The benefit of being in an alpha group is getting information about new NFTs first and getting in on pre-sale/whitelist before everyone else.
There are some entry barriers to alpha groups, but if you're active in the NFT community, you'll eventually bump into, be invited to, or form one.
- A whitelist spot is awarded to members of an NFT community who are the most active and engaged.
This participation reward is the most democratic. To get a chance, collectors must work hard and play to their strengths.
Whitelisting participation examples:
- Raffle, games and contest: NFT Community raffles, games, and contests. To get a whitelist spot, invite 10 people to X NFT Discord community.
- Fan art: To reward those who add value and grow the community by whitelisting the best fan art and/or artists is only natural.
- Giveaways: Lucky number crypto wallet giveaways promoted by an NFT community. To grow their communities and for lucky collectors, NFT projects often offer free NFT.
- Activate your voice in the NFT Discord Community. Use voice channels to get NFT teams' attention and possibly get whitelisted.
The advantage of whitelists or NFT pre-sales.
Chainalysis's NFT stats quote is the best answer:
“Whitelisting isn’t just some nominal reward — it translates to dramatically better investing results. OpenSea data shows that users who make the whitelist and later sell their newly-minted NFT gain a profit 75.7% of the time, versus just 20.8% for users who do so without being whitelisted. Not only that, but the data suggests it’s nearly impossible to achieve outsized returns on minting purchases without being whitelisted.” Full report here.
Sure, it's not all about cash. However, any NFT collector should feel secure in their investment by owning a piece of a valuable and thriving NFT project. These stats help collectors understand that getting in early on an NFT project (via whitelist or pre-sale) will yield a better and larger return.
The downsides of pre-sales & whitelists for NFT creators.
Pre-sales and whitelist can cause issues for NFT creators and collectors.
NFT flippers
NFT collectors who only want to profit from early minting (pre-sale) or low mint cost (via whitelist). To sell the NFT in a secondary market like Opensea or Solanart, flippers go after the discounted price.
For example, a 1000 Solana NFT collection allows 100 people to mint 1 Solana NFT at 0.25 SOL. The public sale price for the remaining 900 NFTs is 1 SOL. If an NFT collector sells their discounted NFT for 0.5 SOL, the secondary market floor price is below the public mint.
This may deter potential NFT collectors. Furthermore, without a cap in the pre-sale minting phase, flippers can get as many NFTs as possible to sell for a profit, dumping them in secondary markets and driving down the floor price.
Hijacking NFT sites, communities, and pre-sales phase
People try to scam the NFT team and their community by creating oddly similar but fake websites, whitelist links, or NFT's Discord channel.
Established and new NFT projects must be vigilant to always make sure their communities know which are the official links, how a whitelist or pre-sale rules and how the team will contact (or not) community members.
Another way to avoid the scams around the pre-sale phase, NFT projects opt to create a separate mint contract for the whitelisted crypto wallets and then another for the public sale phase.
Scam NFT projects
We've seen a lot of mid-mint or post-launch rug pulls, indicating that some bad NFT projects are trying to scam NFT communities and marketplaces for quick profit. What happened to Magic Eden's launchpad recently will help you understand the scam.
We discussed the benefits and drawbacks of NFT pre-sales and whitelists for both projects and collectors.
Finally, some practical tools and tips for finding new NFTs 👇
Tools & resources to find new NFT on pre-sale or to get on a whitelist:
In order to never miss an update, important pre-sale dates, or a giveaway, create a Tweetdeck or Tweeten Twitter dashboard with hyped NFT project pages, hashtags ( #NFTGiveaways , #NFTCommunity), or big NFT influencers.
Search for upcoming NFT launches that have been vetted by the marketplace and try to get whitelisted before the public launch.
Save-timing discovery platforms like sealaunch.xyz for NFT pre-sales and upcoming launches. How can we help 100x NFT collectors get projects? A project's official social media links, description, pre-sale or public sale dates, price and supply. We're also working with Dune on NFT data analysis to help NFT collectors make better decisions.
Don't invest what you can't afford to lose because a) the project may fail or become rugged. Find NFTs projects that you want to be a part of and support.
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middlemarch.eth
3 years ago
ERC721R: A new ERC721 contract for random minting so people don’t snipe all the rares!
That is, how to snipe all the rares without using ERC721R!
Introduction: Blessed and Lucky
Mphers was the first mfers derivative, and as a Phunks derivative, I wanted one.
I wanted an alien. And there are only 8 in the 6,969 collection. I got one!
In case it wasn't clear from the tweet, I meant that I was lucky to have figured out how to 100% guarantee I'd get an alien without any extra luck.
Read on to find out how I did it, how you can too, and how developers can avoid it!
How to make rare NFTs without luck.
# How to mint rare NFTs without needing luck
The key to minting a rare NFT is knowing the token's id ahead of time.
For example, once I knew my alien was #4002, I simply refreshed the mint page until #3992 was minted, and then mint 10 mphers.
How did I know #4002 was extraterrestrial? Let's go back.
First, go to the mpher contract's Etherscan page and look up the tokenURI of a previously issued token, token #1:
As you can see, mphers creates metadata URIs by combining the token id and an IPFS hash.
This method gives you the collection's provenance in every URI, and while that URI can be changed, it affects everyone and is public.
Consider a token URI without a provenance hash, like https://mphers.art/api?tokenId=1.
As a collector, you couldn't be sure the devs weren't changing #1's metadata at will.
The API allows you to specify “if #4002 has not been minted, do not show any information about it”, whereas IPFS does not allow this.
It's possible to look up the metadata of any token, whether or not it's been minted.
Simply replace the trailing “1” with your desired id.
Mpher #4002
These files contain all the information about the mpher with the specified id. For my alien, we simply search all metadata files for the string “alien mpher.”
Take a look at the 6,969 meta-data files I'm using OpenSea's IPFS gateway, but you could use ipfs.io or something else.
Use curl to download ten files at once. Downloading thousands of files quickly can lead to duplicates or errors. But with a little tweaking, you should be able to get everything (and dupes are fine for our purposes).
Now that you have everything in one place, grep for aliens:
The numbers are the file names that contain “alien mpher” and thus the aliens' ids.
The entire process takes under ten minutes. This technique works on many NFTs currently minting.
In practice, manually minting at the right time to get the alien is difficult, especially when tokens mint quickly. Then write a bot to poll totalSupply() every second and submit the mint transaction at the exact right time.
You could even look for the token you need in the mempool before it is minted, and get your mint into the same block!
However, in my experience, the “big” approach wins 95% of the time—but not 100%.
“Am I being set up all along?”
Is a question you might ask yourself if you're new to this.
It's disheartening to think you had no chance of minting anything that someone else wanted.
But, did you have no opportunity? You had an equal chance as everyone else!
Take me, for instance: I figured this out using open-source tools and free public information. Anyone can do this, and not understanding how a contract works before minting will lead to much worse issues.
The mpher mint was fair.
While a fair game, “snipe the alien” may not have been everyone's cup of tea.
People may have had more fun playing the “mint lottery” where tokens were distributed at random and no one could gain an advantage over someone simply clicking the “mint” button.
How might we proceed?
Minting For Fashion Hats Punks, I wanted to create a random minting experience without sacrificing fairness. In my opinion, a predictable mint beats an unfair one. Above all, participants must be equal.
Sadly, the most common method of creating a random experience—the post-mint “reveal”—is deeply unfair. It works as follows:
- During the mint, token metadata is unavailable. Instead, tokenURI() returns a blank JSON file for each id.
- An IPFS hash is updated once all tokens are minted.
- You can't tell how the contract owner chose which token ids got which metadata, so it appears random.
Because they alone decide who gets what, the person setting the metadata clearly has a huge unfair advantage over the people minting. Unlike the mpher mint, you have no chance of winning here.
But what if it's a well-known, trusted, doxxed dev team? Are reveals okay here?
No! No one should be trusted with such power. Even if someone isn't consciously trying to cheat, they have unconscious biases. They might also make a mistake and not realize it until it's too late, for example.
You should also not trust yourself. Imagine doing a reveal, thinking you did it correctly (nothing is 100%! ), and getting the rarest NFT. Isn't that a tad odd Do you think you deserve it? An NFT developer like myself would hate to be in this situation.
Reveals are bad*
UNLESS they are done without trust, meaning everyone can verify their fairness without relying on the developers (which you should never do).
An on-chain reveal powered by randomness that is verifiably outside of anyone's control is the most common way to achieve a trustless reveal (e.g., through Chainlink).
Tubby Cats did an excellent job on this reveal, and I highly recommend their contract and launch reflections. Their reveal was also cool because it was progressive—you didn't have to wait until the end of the mint to find out.
In his post-launch reflections, @DefiLlama stated that he made the contract as trustless as possible, removing as much trust as possible from the team.
In my opinion, everyone should know the rules of the game and trust that they will not be changed mid-stream, while trust minimization is critical because smart contracts were designed to reduce trust (and it makes it impossible to hack even if the team is compromised). This was a huge mistake because it limited our flexibility and our ability to correct mistakes.
And @DefiLlama is a superstar developer. Imagine how much stress maximizing trustlessness will cause you!
That leaves me with a bad solution that works in 99 percent of cases and is much easier to implement: random token assignments.
Introducing ERC721R: A fully compliant IERC721 implementation that picks token ids at random.
ERC721R implements the opposite of a reveal: we mint token ids randomly and assign metadata deterministically.
This allows us to reveal all metadata prior to minting while reducing snipe chances.
Then import the contract and use this code:
What is ERC721R and how does it work
First, a disclaimer: ERC721R isn't truly random. In this sense, it creates the same “game” as the mpher situation, where minters compete to exploit the mint. However, ERC721R is a much more difficult game.
To game ERC721R, you need to be able to predict a hash value using these inputs:
This is impossible for a normal person because it requires knowledge of the block timestamp of your mint, which you do not have.
To do this, a miner must set the timestamp to a value in the future, and whatever they do is dependent on the previous block's hash, which expires in about ten seconds when the next block is mined.
This pseudo-randomness is “good enough,” but if big money is involved, it will be gamed. Of course, the system it replaces—predictable minting—can be manipulated.
The token id is chosen in a clever implementation of the Fisher–Yates shuffle algorithm that I copied from CryptoPhunksV2.
Consider first the naive solution: (a 10,000 item collection is assumed):
- Make an array with 0–9999.
- To create a token, pick a random item from the array and use that as the token's id.
- Remove that value from the array and shorten it by one so that every index corresponds to an available token id.
This works, but it uses too much gas because changing an array's length and storing a large array of non-zero values is expensive.
How do we avoid them both? What if we started with a cheap 10,000-zero array? Let's assign an id to each index in that array.
Assume we pick index #6500 at random—#6500 is our token id, and we replace the 0 with a 1.
But what if we chose #6500 again? A 1 would indicate #6500 was taken, but then what? We can't just "roll again" because gas will be unpredictable and high, especially later mints.
This allows us to pick a token id 100% of the time without having to keep a separate list. Here's how it works:
- Make a 10,000 0 array.
- Create a 10,000 uint numAvailableTokens.
- Pick a number between 0 and numAvailableTokens. -1
- Think of #6500—look at index #6500. If it's 0, the next token id is #6500. If not, the value at index #6500 is your next token id (weird!)
- Examine the array's last value, numAvailableTokens — 1. If it's 0, move the value at #6500 to the end of the array (#9999 if it's the first token). If the array's last value is not zero, update index #6500 to store it.
- numAvailableTokens is decreased by 1.
- Repeat 3–6 for the next token id.
So there you go! The array stays the same size, but we can choose an available id reliably. The Solidity code is as follows:
Unfortunately, this algorithm uses more gas than the leading sequential mint solution, ERC721A.
This is most noticeable when minting multiple tokens in one transaction—a 10 token mint on ERC721R costs 5x more than on ERC721A. That said, ERC721A has been optimized much further than ERC721R so there is probably room for improvement.
Conclusion
Listed below are your options:
- ERC721A: Minters pay lower gas but must spend time and energy devising and executing a competitive minting strategy or be comfortable with worse minting results.
- ERC721R: Higher gas, but the easy minting strategy of just clicking the button is optimal in all but the most extreme cases. If miners game ERC721R it’s the worst of both worlds: higher gas and a ton of work to compete.
- ERC721A + standard reveal: Low gas, but not verifiably fair. Please do not do this!
- ERC721A + trustless reveal: The best solution if done correctly, highly-challenging for dev, potential for difficult-to-correct errors.
Did I miss something? Comment or tweet me @dumbnamenumbers.
Check out the code on GitHub to learn more! Pull requests are welcome—I'm sure I've missed many gas-saving opportunities.
Thanks!
Read the original post here

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.
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Maria Stepanova
3 years ago
How Elon Musk Picks Things Up Quicker Than Anyone Else
Adopt Elon Musk's learning strategy to succeed.
Medium writers rank first and second when you Google “Elon Musk's learning approach”.
My article idea seems unoriginal. Lol
Musk is brilliant.
No doubt here.
His name connotes success and intelligence.
He knows rocket science, engineering, AI, and solar power.
Musk is a Unicorn, but his skills aren't special.
How does he manage it?
Elon Musk has two learning rules that anyone may use.
You can apply these rules and become anyone you want.
You can become a rocket scientist or a surgeon. If you want, of course.
The learning process is key.
Make sure you are creating a Tree of Knowledge according to Rule #1.
Musk told Reddit how he learns:
“It is important to view knowledge as sort of a semantic tree — make sure you understand the fundamental principles, i.e. the trunk and big branches, before you get into the leaves/details or there is nothing for them to hang onto.”
Musk understands the essential ideas and mental models of each of his business sectors.
He starts with the tree's trunk, making sure he learns the basics before going on to branches and leaves.
We often act otherwise. We memorize small details without understanding how they relate to the whole. Our minds are stuffed with useless data.
Cramming isn't learning.
Start with the basics to learn faster. Before diving into minutiae, grasp the big picture.
Rule #2: You can't connect what you can't remember.
Elon Musk transformed industries this way. As his expertise grew, he connected branches and leaves from different trees.
Musk read two books a day as a child. He didn't specialize like most people. He gained from his multidisciplinary education. It helped him stand out and develop billion-dollar firms.
He gained skills in several domains and began connecting them. World-class performances resulted.
Most of us never learn the basics and only collect knowledge. We never really comprehend information, thus it's hard to apply it.
Learn the basics initially to maximize your chances of success. Then start learning.
Learn across fields and connect them.
This method enabled Elon Musk to enter and revolutionize a century-old industry.

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

Asher Umerie
3 years ago
What is Bionic Reading?
Senses help us navigate a complicated world. They shape our worldview - how we hear, smell, feel, and taste. People claim a sixth sense, an intuitive capacity that extends perception.
Our brain is a half-pool of grey and white matter that stores data from our senses. Brains provide us context, so zombies' obsession makes sense.
Bionic reading uses the brain's visual information and context to simplify text comprehension.
Stay with me.
What is Bionic Reading?
Bionic reading is a software application established by Swiss typographic designer Renato Casutt. The term honors the brain (bio) and technology's collaboration to better text comprehension.
The image above shows two similar paragraphs with bionic reading.
Notice anything yet?
This Twitter user did.
I did too...
Image text describes bionic reading-
New method to aid reading by using artificial fixation points. The reader focuses on the highlighted starting letters, and the brain completes the word.
How is Bionic Reading possible?
Do you remember seeing social media posts asking you to stare at a black dot for 30 seconds (or more)? You blink and see an after-image on your wall.
Our brains are skilled at identifying patterns and'seeing' familiar objects, therefore optical illusions are conceivable.
Brain and sight collaborate well. Text comprehension proves it.
Considering evolutionary patterns, humans' understanding skills may be cosmic luck.
Scientists don't know why people can read and write, but they do know what reading does to the brain.
One portion of your brain recognizes words, while another analyzes their meaning. Fixation, saccade, and linguistic transparency/opacity aid.
Let's explain some terms.
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Fixation is how the eyes move when reading. It's where you look. If the eyes fixate less, a reader can read quicker. [Eye fixation is a physiological process](Eye fixation is a naturally occurring physiological process) impacted by the reader's vocabulary, vision span, and text familiarity.
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Saccade - Pause and look around. That's a saccade. Rapid eye movements that alter the place of fixation, as reading text or looking around a room. They can happen willingly (when you choose) or instinctively, even when your eyes are fixed.
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Linguistic transparency and opacity analyze how well a composite word or phrase may be deduced from its constituents.
The Bionic reading website compares these tools.
Text highlights lead the eye. Fixation, saccade, and opacity can transfer visual stimuli to text, changing typeface.
## Final Thoughts on Bionic Reading
I'm excited about how this could influence my long-term assimilation and productivity.
This technology is still in development, with prototypes working on only a few apps. Like any new tech, it will be criticized.
I'll be watching Bionic Reading closely. Comment on it!
