Challenge #420: The Sierpinski Triangle

Hi Maveryx,

A solution to last week’s challenge can be found here.

This week, we are diving deep into the realms of Math and Spatial tools by tackling the creation of Sierpinski’s triangle fractal. This challenge, designed by Roland van Leeuwen @RWvanLeeuwen, is an Expert-level task. If you are preparing for certification and plan to attempt an exam during Inspire, it is an excellent opportunity to hone your skills. Thank you, Roland, for crafting this challenge!

What is a Sierpinski triangle?

A Sierpinski triangle is a fractal shape composed of smaller triangles, each a scaled-down replica of the whole. It is created by repeatedly dividing an equilateral triangle into smaller triangles and removing the middle triangle at each iteration. This process results in a geometric pattern that exhibits self-similarity at different scales, forming a visually striking and intricate triangle-based fractal.

(This definition is sourced from https://en.wikipedia.org/wiki/Sierpi%C5%84ski_triangle.)

The provided input consists of the of latitude, longitude, and corners a, b, and c. (The corners are used to determine each point of the triangle.) Your output triangle will look like this:

For this challenge, we are providing additional guidance to simplify the tasks and help you build your Sierpinski’s triangle.

To construct a fractal triangle, follow these steps:

Find the corners (points a, b, and c), and any random point within the triangle as a starting point.
Select one of the corners of the triangle and form a line from the point to the corner.
The center point of the line created is the one we will iterate with.
Using the center point of the created line, pick another random corner, draw a line, and find its center.
Repeat steps 2 to 4 for 100 iterations.
Map all points as green diamonds and the first random centroid as black.

By repeating these steps, a fractal should appear in the shape of Sierpinski's triangle!

If you need a refresher on how to build an iterative macro or create spatial objects, review these lessons in Academy to gear up:

Good luck!

AYX-Maveryx_Micro Identity-Maveryx Academy_wordmark.png

Challenge_420_start_file.yxzp

Challenge_420_solution_file.yxzp

Macros

Spatial Analysis

Expert

Difficult

Accepted answers

All comments

Kenda

This definitely seems like a @patrick_digan math problem

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I went for 10k iterations and triangle shapes rather than diamonds

Challenge_420_ks.yxzp

mmontgomery

Oof, this was a doozy. More in spoiler

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It took me a while to understand what to do. I tweaked step 4 by taking the centroid of a random point and a random vertex. The new centroid would be my starting point, which would then map to a random vertex and so on.

Also, I iterated it 2000 times for a more holistic picture.

c420.yxzp

ScottLewis

That was fascinating. Having a hard time with why the algorithm works from any point but it sure does.
Needed a lot more than 100 iterations to get something filled out. Wonder if that meant 100, from each existing point each time.

I just kicked it up to 10k and the pattern emerges.

Challenge420MacroScott.yxmc

Challenge_420_Scott.yxmd

gawa

I like math, I like this challenge.

When it comes to a challenge associated with Iteration Macro, it's very quiet here. It implies that Iterative operation in Alteryx would be one of the hurdle for the most users.

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I could see the challenging tasks this week. First, we need to create Iterative Macro. Second, we need to use Spatial functions in Formula tool.
To create a random point, ST_RandomPoint([obj]) is very handy.

Some observation:
(1) We need at least 1,000 iterations to get a vivid view, we are instructed to go with 100 iterations though. If your computer allows, go with 100k.

100

10k

(2) Point created during the first 2 iterations sometimes fall outside the triangle(sometime not). I omitted the record from first 2 iterations.

Challenge_420_gawa.yxzp

balajilolla2

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Solution Attached

Screenshot 2024-04-01 205741.jpg

Challenge_420_start_file.yxzp

Qiu

Math is really magical.

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100 Iterations

30K iterations

Challenge_420_start_file-Qiu.yxzp

JamesCharnley

Great challenge! I posted a similar challenge before on Databasyx.com so I've gone a bit rogue and have a couple of other ways to create the Sierpinski triangle in Alteryx!

One way is similar to this challenge with using midpoints but cutting out shapes instead using spatial process. The other is to colour in every even number in a Pascal's Triangle!

Edit: Pictures in below post. Failed to upload and couldn't edit them in!

JamesCharnley

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Apologies for double post but pictures failed to upload in my previous post and can't edit them in still!

RolandSchubert

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Challenge_420_solution_rsc.yxzp

ed_hayter

Great challenge to learn that ST_RandomPoint() is a function that can be used in the formula tool.

Challenge_420_start_file.yxzp

sergejs_kutkovics

My try This was a great challenge, thank you, @RWvanLeeuwen!

I also didn't know about ST_RandomPoint function.

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Workflow

Macro

Output

Challenge_420_SK.yxzp

alexnajm

Upped it to 10K iterations to make it show better - what an interesting phenomenon!

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Challenge_420_start_file_Alex Abi-Najm.yxzp

RWvanLeeuwen

I love it when math becomes beautiful

Here's my take:

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Workflow:

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Macro:

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Weekly Challenge 420 - RWvL - Sierpinskis Triangle.yxzp

JasperMB

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Challenge_420_solution_JB.yxmd

Pilsner

Great challenge, I love the mathematical element.

My solution:

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The container contains the tools I had in the macro:

The picture was run with 10k iterations:

420 results.png

420 (Fractal Triangle).yxzp

olga_strubbe

Thank you, @RWvanLeeuwen!

This was a unique and interesting challenge. I did not know how the Sierpinski triangle is built until now. Thanks for sharing.

I leveraged your approach in building my solution.

Output with 1,000 new data points:

Workflow:

Iterative macro:

olga_strubbe

sorry, I forgot to include the workflow & macro, here they are.

Challenge_420_solution OS.yxmd

challenge 420_iterative macro.yxmc

nSpire

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Challenge_420_nSpire.yxzp

AkimasaKajitani

My solution. I created it using my own approach focused on creating the Sierpinski Triangle without following the start file text.

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I don't use the random function. I created three triangles from one triangle and processed them recursively.

スクリーンショット 2024-04-07 212239.png

This sample output is made from repeating 10 times.

Challenge_420_start_file_AK.yxzp

abe_ibanez

This was really fun.

Thank you for putting it together!

Here is my solution:

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I ran the iteration 5,000 times to get a fuller image.

Challenge_420_start_file.yxzp

Yoshiro_Fujimori

Skipped the part to get the start point randomly.

I would try using ST_RandomPoint() function when I have time.

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Workflow

Macro

Output (1,000 iterations)

Challenge_420_YFujimori.yxzp

ahsanaali

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Challenge_421_solution_AAA.yxzp

gawa

@Qiu gave me a bonus assignment:Sierpiński carpet😁 It is a beauty of math.

https://en.wikipedia.org/wiki/Sierpi%C5%84ski_carpet

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Sierpinski Carpet.yxzp

DaisukeTsuchiya

Interesting!

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Challenge_420_start_file_DT.yxzp

lynnesonney

study study study!

I overthought this one

iterate_420.yxmc

Challenge_420_start_file.yxmd

bgreedy

Feel like I've stumbled at the last hardly, I've managed to get all the plots for the triangle but when I try to render with the report map I get a massive square, the preview matches the expected output so I'm a bit stumped, if anyone can spot where I went wrong.