Round Square Combinations are Surprising

November 30, 2023 · Posted in 1st Principles, science · Comment 

All four of these solids are Round Squares. I thought I had a general rule for calculating the combinations when I found the first two but then I found another and another. Is there a general formula for all the possible combinations. What are the parameters? Numerically there could be 6 but I do not think geometrically 6 are possible. This gets more confusing for Round Triangles or Square Triangles.

Cylinder is the obvious round square.


Cones are round triangle. Two triangles for a square. Two cones form a round square.

Union of two cones

Combinations of 3D shapes are either unions or intersections. The intersection of a cylinder and a wedge makes a Round Square Triangle. The union of two half height Round Square Triangles form a Round Square.

Intersection of a cylinder and a diamond extrusion.

Intersection of two cylinders rotates 90 degrees from each other leaves a square and two rounded sides for the Fourth Round Square.

Intersection of cylinders rotated 90 degrees from each other,

I found the first two using pencil and paper. The third with FreeCAD thinking in a subtractive then additive fashion. The fourth I found using OpenSCAD. The different approaches enabled finding the alternatives. I would not have found the fourth type using pencil and paper, the CAD software showed me what I described.

How would you go about finding a proof for a general formula?

Day 43 – Jamming in Bangkok

April 6, 2019 · Posted in 1st Principles, mindset · Comment 

Today I rode my bicycle to EQ Studios to jam with my friend Sam & his friend Brad. Music improvisation is possible because of first principles. We followed a few basic rules so we can spontaneously make music without planning. The key point of first principles is anything that can be made or done can be assembled from a very small set of fundamental components and rules repeated over & over. This is the idea of fractals.

To add to the improvisational aspect I took the video of riding my bicycle to the studio and speed it up to match the length of the music. You can see and hear all of this because the video is composed of pixels, tiny dots of 3 colors arranged in a grid so you see the picture. You hear the music because it is also composed of a string of numbers or digits representing the sound level in very small time slices so when sent as electric impulses to the speaker it vibrates reproducing the original sound.

Riding a bicycle is also done following very simple rules. You balance by making small adjustments of the handle bar left or right. You also make very small adjustments of your body weight to help keep your balance. You move forward by pushing down on the peddle, first the right then the left. A bunch of small actions repeated over and over to achieve a fluid result.

All of the vehicles on the street in Bangkok are also improvising. They watch each other and adjust their current and future actions to spontaneously form coordinated motions to get where each wants without hitting anything.

When you look you can see the 1st Principles operating behind everything.

Day 34 in Bangkok – Measuring Fat, Products, & Desires

March 20, 2019 · Posted in innovation · Comment 

Day 34 of First Principles in Bangkok is a product review of a Xiaomi Mi Fit Smart scale as an example of measuring fat to explain my startup app QuickMatch that measures “fuzzy” desires. I cover the entire process of choosing the product, receiving it, setup, & using it. There are many meta lessons in this video.