# The Lorenz Collection

*Chaos Theory* is excited to announce our first collection: *The Lorenz Collection.*

*Background*

*Background*

Chaos Theory is a branch of mathematics centered around “the Butterfly effect,” where a small, often seemingly insignificant change in the initial conditions results in unpredictable and often drastic changes.

The Lorenz Collection focuses on the genesis example of Chaos Theory: The Lorenz Attractor.

The Lorenz Attractor is a model that calculates the flow of fluid over time under multiple varying conditions. As the system has multiple independent variables that depend on each other, it is impossible to predict a point accurately in the future without first calculating points between a given initial point and the desired point (the more points calculated in between, the more accurate the prediction), or in other words, the system exhibits “chaotic behavior.” The shape of this attractor often resembles that of a butterfly. A more rigorous technical explanation can be found here.

The simplified system of ordinary differential equations:

This project uses σ = 10, β = 8 / 3, and ρ = 28, as did Edward Lorenz when showcasing the beauty of chaotic behavior.

# Parameters

**Initial Condition **—** **The initial point for the Lorenz system of equations and the crux of this collection. As each generation guarantees a completely unique initial point, each resulting solution is unpredictably unique.

Distribution: Each component (x0, y0, z0) ranges from -3.0 to 3.0, with all values being equally likely.

**Palate **—** **Selects the sequential colormap that illustrates the Lorenz, with the starting points of the Lorenz more bright and the ending points more saturated. The bank of sequential colormaps is depicted below:

Distribution: All colors are equally likely.

**Azimuth, Altitude, and Radius** — Describes the position of the camera about a sphere encompassing the Lorenz. In static form, this is the frame of reference used for the 2D representation. In animated form, this position serves as the starting point from which the camera orbits about the sphere.

Distribution: Azimuth and Altitude range from 0 to 360 degrees, with all values being equally likely. All Lorenz’s use a Radius of 9.

**Theta, Phi, and Speed** — Relevant in animated form, details the orbit about the sphere. Theta and Phi actually refer to dΘ/dt and dΦ/dt, where every given time interval (dictated by speed), the camera is moved a given amount in Θ and Φ about the sphere. The set of possible values are (Θ, Φ) = {1 1, 1 -1, -1 1, -1 -1, 1 0, -1 0, 0 1, 0 -1}, as these ratios allow for the largest possible orbit about the sphere while maintaining a circular orbit.

Distribution: the values for theta and phi are equally likely. Speed is evenly distributed between slow, normal, and fast.

**Lines** — The continuity of the lines within the system of equations, ranging from solid to sparse, with a significant bias towards generating more continuous lines.

Distribution: 60% Solid, 25% Abundant, 10% Woven, and 5% Sparse.

**Points**—The number of points drawn for the Lorenz. All Lorenz’s use 100,000 points.

**Drawn **— True or False. Relevant in animated form, shows the Lorenz being drawn from the initial point to the number of points specified in the points parameter.

Distribution: 20% True, 80% False

# Transform

The Lorenz Collection is the first semi-mutable NFT collection on Solana, as the parameters defining the Lorenz are immutable, but the visual representation is mutable. By default, every NFT includes a static 2D visual representation of the Lorenz described by its parameters. Owners will be able to convert a static representation of their Lorenz into an animated one, or vice versa.

As the cost of each transform scales with a bonding curve, the number of transformations will be capped by demand. The price will rise according to the bonding curve every 20 transformations.

# The Drop

**Supply: 6720**

This number comes from the Lorenz constants used both in this project and in Edward Lorenz’s demonstrations. σβρ = 10 * (8 * 3) * 28 = 6720.

**Presale (25%): 15th October 5PM UTC — Mint Price 0.75 SOL**

A quarter of the supply, or 1680, will be sold during the presale.

After the time specified above, to mint, visit chaostheorynft.com and press mint. Your Solana address must be on the whitelist to mint during the presale.

The whitelist form is available here.

**Main Launch: Late October — Mint Price 1 SOL**

The remaining 5040 Lorenz’s will be sold at launch. Exact date TBD. During the launch, visit chaostheorynft.com and press mint.

**Royalties**

There is a 5% royalty on all sales going towards a creator-controlled community fund. This fund will be 100% transparent, and is designed to advance the brand of Chaos Theory and bring value to its collections while allowing for community input.

*Follow us on our social media for updates regarding the project:*

**Website:** chaostheorynft.com

**Twitter: **twitter.com/ChaosTheoryNFTs

**Discord:** discord.gg/e6AHTN3dcH