Sierpinski Pyramid

This began with the aim of making 3D printing an enrichment to the maths curriculum.

While searching I came across this article, 3D printed maths art: 10+ examples. Their number one example was the Sierpinski Pyramid, so lets run with that.

You can find it on thingiverse, Sierpinski pyramid. The animated gif shows how it is made:

The Summary at thingiverse says:

It has the astonishing property that the horizontal cross sections are simply connected and change continuously with height, so can be printed perfectly with spiral mode (or single-walled without infill).

This fractal is half of the octahedron flake (or Sierpinski octahedron). Every triangular face is a Sierpinski triangle, and the base of the pyramid is (approximately) a space-filling curve called the Sierpiński curve. Figures like the Vicsek fractal also exist in this model.

I must admit that I didn’t understand “spiral mode” and went ahead just by setting infill to 0%. Fortunately, this worked well. Later on I went back and looked up “spiral mode” and I now understand it.

The image at the start shows the 3 different sized pyramids I made on my Prusa MINI+. The small version took roughly 1 hour; the medium version roughly 5 hours; and the large version required 29 hours!!

Here are some progress images of the larger version print:

Further Sierpinski Pyramid possibilities (thingiverse remixes):

Fractal Pyramid Lamp and Base I really loved the look of the fractal pyramid and wanted to turn it into a lamp so i designed a base for it that will house a led strip and print the pyramid with transparent filament to turn it into a lamp. I imagine a glow in the dark filament would work pretty well too. Color of the base is whatever you choose, I recommend using some variation of Silk filament for its reflective properties.

How was the Sierpinski Pyramid made?

Years ago I learnt recursion from Brian Harvey's Logo books. Here is a link to a recursive 2D Sierpinski triangle on the Snap site. Brian is one of the Snap developers.

I just did a simpler one myself (here), without colour or pen size variation. Here's the code:

The making of the 3D version is not explained on thingiverse. The best link I've found so far for the 3D making is this article / poster, Visualizing Fractals Using 3D Printing

Some Further research:
Wolfram language: Sierpinski Mesh

Math Art: Search in thingiverse for the label “math art” (793 things) or “mathematical art” (143 things) and you’ll find lots of possibilities.

The Inevitable: Introduction

Kelly, Kevin. The Inevitable: Understanding the 12 Technological Forces that will Shape our Future (2017)

"... the computer age did not really start until ... computers merged with the telephone. Stand alone computers were inadequate ...

There is a bias in the nature of technology that tilts it in certain directions and not others ... The roots of the digital world are anchored in the physical needs and natural tendencies of bits, information and networks ... Their inevitability stems from their basic physics"

KK initially found computers boring. This changed for him when they became connected and a gigantic world opened up at the other end of the phone line.

Initially, I didn't see the point of computers either. But this changed for me after I read Mindstorms by Seymour Papert. His idea was that a programming language, logo (a simplied version of Lisp), could be used to make maths far more interesting. Turtle geomety. This did arise from the natural tendency of bits, their ability to render variations in geometrical graphics far more quickly than by pencil. This happened before networks so I disagree with KK on that point.

He means inevitable in the sense that new technologies will inevitably create certain things although their form may vary. Some form of internet was inevitable (since humans in general want to communicate) but the exact form of an internet (a network of networks) is not inevitable (public, private, national differences exist, certain things are censored by some countries etc.)

Massive copying is here to to stay. Massive tracking and total surveillance is here to stay. ... We can't stop artificial intelligence and robots ...

Of course, he argues for regulation throughout the book. More on "total surveillance" in Ch 10 Tracking, with reference to Ed Snowden.

"All is flux. Nothing is finished ... This never ending change is the pivotal axis of the modern world.... Our greatest invention in the past 200 years was not a particular gadget or tool but the invention of the scientific process itself ... the process generated a million new products over the centuries ... processes trump products

My only caveat here is that KK talks about the "scientific process" without acknowledging that there is no real agreement about what that really is. eg. refer Andrew Pickering's "The Mangle of Practice: Time, Agency & Science", one of many books that interrogates the scientific method.

We are moving away from a world of fixed nouns and towards a world of fluid verbs ... I've waded through the myriad technological forces erupting into the present and I've sorted their change into 12 verbs, such as accessing, tracking, and sharing. To be more accurate, these are not just verbs but present participles, the grammatical form that conveys continuous action.

I think this approach is magnificent. The theme of the book is continuous flux and he discusses the particular technologies in the framework of these action verbs. This is much more interesting than just describing the technologies.

"... digital technologies (computers, internet, apps) favour cheap ubiquitous duplication ... (this) bias is independent of nationality, economic momentum, or human desire, and it steers the technology towards social ubiquity ..."

I looked up the meaning of social ubiquity: presence everywhere or in many places especially simultaneously : omnipresence