Perfectly accurate parametric Coffee Spoon

This started a long time ago. The problem I wanted to solve was to make a coffee spoon (Kaffeelot) to fill the right amount of ground coffee into the machine.

I had one if these iconic conical spoons with a rounded cap. My design had to replicate this form.

However: I wanted to make it parametric so I can set a target volume and define the basic characteristics of the form like the relation between the base diameter and the height and the radius of the rounded tip.

I derived the equation to get the volume from the dimensions using trigonometric functions. Later I discovered that this was already done and on the Internet for all to see:

https://rechneronline.de/pi/rounded-cone.php

I copy+pasted the math but I guess you are better off viewing it using the link above. Also check out the PDF which includes the conversation with the AI.

α = arccos [ ( R - r ) / √(h² + ( R - r )²) ]

i = √(r² / [ 2 / ( 1 - cos α ) - 1 ])

h = i + j

A = ( R + r ) π √(( R - r )² + j²) + πR² + π * ( r² + i² )

V = π/3 { j ( R² + Rr + r² ) + i² [ 3 ( r² + i² ) / (2i) - i ] }

I put in the factors for r and j and tried to solve it. I failed because the trigonometric functions got in the way and my math skills weren't strong enough to solve this.

Then, almost 3 years after I started this project I read an article about AI solving mathematical conjectures. This gave me the idea to try it again using AI.

And it worked.

I double checked the result with the volume calculator in the link above and it checks out.

The AI found a way to completely avoid trigonometric functions thus making it much easier to solve for the base radius. It also wrote the OpenSCAD code. All I did was attaching a handle to it.

There's a PDF with the complete conversation with the AI in the download section.

The handle has the target volume labeled onto it. It also considers the number of cups. E.g. it reads "2x 15ml" instead of "30ml".