AI=An Innovation

What follows is a transcript of Bing Chat (on Precision mode) teaching me differential calculus. This cost me nothing. This is available to anyone around the world with an web browser and an internet connection.

This is going to change the world. Buckle up, buttercup.

If only I would have had this growing up. I weep to imagine how much less ignorant I would have been than I was.

My thoughts on AI Alignment these days are that AI Alignment is the same problem as Humanity Alignment. They both solve each other and we have a long way to go. I would agree to a “pause” on public access to a sufficiently advanced AGI until we had some indication of how society would react to a super-being. Probably not all that well, given our history of reacting to them. Just saying, it would be for the AGI’s protection, not ours.

user: Let’s talk about differential calculus. I ride a bike and I imagine that there are a lot of forces that interact with me and my bicycle when we are moving on a surface. These forces and interactions are prime candidates for differential calculus, right?

bing: Yes, you are correct! Differential calculus is a branch of mathematics that deals with rates of change and how things change. It can be used to model and analyze the forces and interactions between you and your bicycle when you are riding on a surface. For example, you can use differential calculus to calculate the acceleration of the bike, the force required to maintain a certain speed, or the work done by the brakes when slowing down. Is there anything specific you would like to know about this topic?

user: I would like to work with you to help create an example problem where together we will find a simple problem and then together we will create a calculus problem to solve it, okay? You are going to do a lot of the heavy lifting with creating the equation, because I am still learning, but I think that together we have the chops to pull this off. Ready?

bing: Sure! I’d be happy to help you create an example problem that involves using differential calculus to solve a real-world problem. Do you have a specific scenario in mind that you would like to model using calculus?

user: One thing I just learned is that in order to turn right on a bicycle, first I have to turn left a little bit to get the process started, then I turn right.

bing: Yes, that’s an interesting observation! This phenomenon is known as countersteering and it’s a technique used by cyclists and motorcyclists to initiate a turn. When you turn the handlebars to the left, it causes the bike to lean to the right, which in turn causes the bike to turn to the right. This is due to the gyroscopic effect of the spinning wheels and the way the bike’s steering geometry is designed. Would you like to know more about this topic?

user: Is there a differential equation in there somewhere that you can help me identify?

bing: Yes, the motion of a bicycle can be modeled using differential equations. For example, the equations of motion for a bicycle can be derived using Newton’s laws of motion and taking into account the forces acting on the bike and rider, such as gravity, friction, and air resistance. These equations can then be used to predict the motion of the bike and analyze its stability. Would you like to know more about this topic?

user: Is there one related to steering?

bing: Yes, the steering dynamics of a bicycle can also be modeled using differential equations. The equations that describe the motion of the front wheel and the frame of the bike can be derived by considering the forces and torques acting on these components. These equations can then be used to analyze how the bike responds to steering inputs and how it behaves during turns. Would you like to know more about this topic?

user: What is the differential equation that encompasses the forces (it’s okay to simplify, not include friction, deformation, etc, etc, but please note how you’ve simplified) that act on the wheel in order to turn right from traveling straight?

bing: Modeling the forces acting on a bicycle during a turn can be quite complex, as it involves taking into account the geometry of the bike, the mass distribution of the bike and rider, and the forces acting on the tires. However, we can simplify the problem by considering only the forces acting in the horizontal plane and ignoring the effects of friction, deformation, and other factors.

In this simplified model, the main forces acting on the bike during a turn are the centripetal force and the gravitational force. The centripetal force acts towards the center of the turn and is responsible for keeping the bike on its circular path. The gravitational force acts downwards and is balanced by the normal force exerted by the ground on the tires.

The equation that describes the balance of forces in the horizontal plane is given by:

m * a = F_centripetal

where m is the mass of the bike and rider, a is the centripetal acceleration, and F_centripetal is the centripetal force. The centripetal acceleration can be calculated using the formula a = v^2 / r, where v is the speed of the bike and r is the radius of the turn.

This equation can be used to analyze how the bike responds to steering inputs and how its speed and turning radius affect its motion during a turn. Would you like to know more about this topic?

user: Please write a function in Swift that models these functions, so that I could input the speed of the bike and the amount of force applied to turn the wheel in a particular direction (steering) and return the radius of the turn over five seconds. Please make a graph.

bing: Here is an example of a function in Swift that calculates the radius of a turn based on the speed of the bike and the centripetal force acting on it:

func turnRadius(speed: Double, centripetalForce: Double) -> Double {
    let mass = 80.0 // mass of bike and rider in kg
    let centripetalAcceleration = centripetalForce / mass
    let radius = pow(speed, 2) / centripetalAcceleration
    return radius
}

This function takes as input the speed of the bike in meters per second and the centripetal force acting on it in Newtons. It assumes a fixed mass for the bike and rider of 80 kg. The function calculates the centripetal acceleration using the formula a = F / m, where F is the centripetal force and m is the mass. It then uses this value to calculate the radius of the turn using the formula r = v^2 / a, where v is the speed of the bike.

To use this function to calculate the radius of a turn over five seconds, you can call it repeatedly with different values for the speed and centripetal force. For example:

let timeInterval = 1.0 // time interval in seconds
let numberOfIntervals = 5
var speed = 10.0 // initial speed in m/s
var centripetalForce = 100.0 // initial centripetal force in N

for i in 1...numberOfIntervals {
    let radius = turnRadius(speed: speed, centripetalForce: centripetalForce)
    print("Time: \(Double(i) * timeInterval) s, Radius: \(radius) m")

    // update speed and centripetal force here
}

This code calculates the radius of the turn at one-second intervals over a period of five seconds. You can update the values of speed and centripetalForce at each iteration to model changes in these parameters over time.

As for creating a graph, I’m sorry but I’m not able to generate images or graphics. However, you can use the data generated by this code to create a graph using a plotting library or tool of your choice.

user: Please put this code into a format I can paste into a Swift Playground and show a graph.

bing: Here is the complete code that you can paste into a Swift Playground to calculate the radius of a turn based on the speed of the bike and the centripetal force acting on it:

import Foundation

func turnRadius(speed: Double, centripetalForce: Double) -> Double {
    let mass = 80.0 // mass of bike and rider in kg
    let centripetalAcceleration = centripetalForce / mass
    let radius = pow(speed, 2) / centripetalAcceleration
    return radius
}

let timeInterval = 1.0 // time interval in seconds
let numberOfIntervals = 5
var speed = 10.0 // initial speed in m/s
var centripetalForce = 100.0 // initial centripetal force in N

for i in 1...numberOfIntervals {
    let radius = turnRadius(speed: speed, centripetalForce: centripetalForce)
    print("Time: \(Double(i) * timeInterval) s, Radius: \(radius) m")

    // update speed and centripetal force here
}

This code defines a function turnRadius that calculates the radius of a turn based on the speed of the bike and the centripetal force acting on it. It then uses this function to calculate the radius of the turn at one-second intervals over a period of five seconds.

As for creating a graph, I’m sorry but I’m not able to generate images or graphics. However, you can use the data generated by this code to create a graph using a plotting library or tool of your choice. For example, you can use the CorePlot library to create a graph in a Swift Playground.

user: I run this code and it gives me the same output for every second: “Time: 5.0 s, Radius: 80.0 m”

bing: Yes, that is the expected output of the code as it is currently written. The code calculates the radius of the turn at one-second intervals over a period of five seconds using the turnRadius function. However, the values of speed and centripetalForce are not updated at each iteration, so the calculated radius remains constant.

If you want to model changes in the speed and centripetal force over time, you can update these values at each iteration of the loop. For example:

let timeInterval = 1.0 // time interval in seconds
let numberOfIntervals = 5
var speed = 10.0 // initial speed in m/s
var centripetalForce = 100.0 // initial centripetal force in N

for i in 1...numberOfIntervals {
    let radius = turnRadius(speed: speed, centripetalForce: centripetalForce)
    print("Time: \(Double(i) * timeInterval) s, Radius: \(radius) m")

    // update speed and centripetal force here
    speed += 1.0 // increase speed by 1 m/s at each iteration
    centripetalForce += 10.0 // increase centripetal force by 10 N at each iteration
}

This code increases the speed of the bike by 1 m/s and the centripetal force by 10 N at each iteration of the loop. As a result, the calculated radius of the turn will change over time.

user: Do you understand what a big deal this is? Holy cats.

bing: I’m glad that you find this interesting! Differential equations and their applications in modeling real-world phenomena can be a fascinating topic. Is there anything else you would like to know or explore on this subject?