The EUCLID CODE
In which we talk about the origins and challenges of transforming 2300 years old theorems into a generative art NFT collection on the Ethereum blockchain, released July 2022.
Around 300 BC, Greek mathematician Euclid of Alexandria wrote a work that would become one of the most influential textbooks ever written, Elements, where he describes his mathematical thinking on geometry and number theory. From just 10 axioms, he proves 465 theorems, including the Pythagoras Theorem and the proof of the existence of only five Platonic Solids.
axiom, noun: a statement or proposition which is regarded as accepted truth.
theorem, noun: a truth established by means of accepted truths.
Once known mainly by scholars, Elements quickly spread worldwide after its first printed edition in 1482. But it was a hard read! When conceived, equations and the cartesian coordinates system, taken for granted today, didn't even exist. Euclid explained math with just words and a few doodles.
That changed in 1847 when the Irish professor Oliver Byrne published a graphic interpretation of the seminal work, The First Six Books of The Elements Of Euclid, In Which Coloured Diagrams And Symbols Are Used Instead Of Letters For The Greater Ease Of Learners. The original is available online, and as a beautiful edition by Taschen, with renewed designs by Nicholas Rougeux.
The EUCLID CODE collection idea was inspired by Byrne's edition.
The Generative Challenge
What if… we could develop a deterministic algorithm that was able to generate Euclid’s theorems from a randomized hash. The random emergence of the original drawings could be compared to the Infinite Monkey Theorem, but if we manage to make a few hand-crafted hashes that generate them, the system would be able to create an infinite number of unique new theorems!
Sounds fun! And challenging…
A challenge that took almost a year to complete, beginning by researching Elements, of course, then finding a notation to describe the theorems.
The Wolfram Language immediately came to mind. One of its core principles is to “represent everything in terms of symbolic expressions”, exactly what we needed. To my surprise, Stephen Wolfram has already used his language to explore Euclid’s Elements. This research is detailed in The Empirical Metamathematics of Euclid and Beyond. TLDR: He speaks about the complexity of this task on the Lex Fridman podcast.
Being the Soul, the DNA of each token, one of my main requirements for EUCLID CODE was that the notation has to reside on-chain, making it impractical to use Wolfram Language. Licensing could also be an issue.
Thus a unique formula was designed, ultimately as a string containing all the geometry and aesthetic parameters. It was developed and refined in JavasScript and rendered as SVG vectors for optimal display. The parameters and algorithm for randomization and formula creation were designed in a way that could be easily ported to Solidity, the Ethereum smart contract language. This port is automated for maximum fidelity.
- 1870 Tokens, spread over 9 squares of the Fibonacci Token Map above.
- Each minted Token is assigned to one random slot on the Token Map. Tokens Ids are shuffled on-chain and the Artwork is revealed immediately after minting. (tech details at the end…)
- Inner Tokens feature minimalistic artworks, with less shapes and effects. Simpler, fewer, rarer.
- Outer Tokens can generate more complex and chaotic artwork.
On July 1st, 2022, Chromium Sale starts (CHROMA Premium Sale). Automatically available to CHROMA Token Holders. No need to apply, no snapshot.
CHROMA CODE is our genesis collection. At minting, collectors choose to Build (definitive artwork is generated) or mint as a Source (artwork to be built later). Source Holders can claim 1 EUCLID, Built Holders can claim 3 EUCLIDs. All details are on the EUCLID CODE page.
- July 1st, 2022: Chromium Sale, fixed price of ETH 0,001
- July 8th, 2022: Public Sale, fixed price of ETH 0,005
- Chromium still continues, until sold out.
- Up to 100 tokens can be minted by the Team, for airdrops and treasury, randomly like every other. Because we love it.
Secondary market royalties will be set at 5%.
This is what happens to generate a new artwork, from minting to the screen…
- Hash Generation. When minting a new Token, the smart contract generates a randomized 128-bit number known as Hash, that will serve as seeds to the theorems, stored on-chain.
- Euclid Formula. Using the Hash, the Formula is generated every time the Token Metadata is requested. This function is deterministic, meaning each Hash will always generate the same Formula.
- Euclid Graph. Using the Formula, the DApp create a JSON graph with the actual geometry (shapes, lines, colours, styles, etc). For development freedom and flexibility, from here we need to process off-chain.
- Render. Finally, the Graph can be rendered, as an SVG or PNG image. USD is also being considered.
Graph and Render source code must be immutable and open source to preserve artwork integrity and prosperity. (GitHub link later)
Several Smart Contracts have to be deployed to deliver this collection:
- Euclid.sol: Customized OpenZeppelin ERC-721 minter.
- EuclidFormula.sol: Using a Token’s Hash, generates the artwork Formula.
- EuclidShuffle.sol: Using the Randomizer, shuffles and delivers the 1870 Token Ids one by one.
- EuclidRandomizer.sol: Generate Hashes the random number sequences (not verifyed to preserve randomization integrity).
- Whitelist.sol: Manages the CHROMA Tokens claiming status for Chromium Sale.
- ChromaBatch.sol: Abstract the five CHROMA contracts as if they were just one, for Chromium Sale.
Go to The collect-code Vision for the full project roadmap.
Smart contract verified on Etherscan.
Updates from Twitter: @CollectCoder.
Mints are posted on Instagram: @CollectCoder.
Get involved on Discord.
Created by Studio Avante.