Category: Fractal


MRI scans of vegetables

I stumbled across the “Inside Insides” blog today which seems to feature various fruits and vegetables as seen by slices through Magnetic Resonance Imaging. Be sure to check out the whole site here

 

Cross Section of a watermelon


Broccoli

Artichoke Cross-section

Mandelbox Zoom

More in the 3D mandelbulb craze thats been sweeping the fractalsphere lately. It all started with the skytopia post on mandelbulbs.

Here is the mandelbulb 3d software used to create the video if you want to play around yourself

Oh and I would turn off the sound… I think the music is horrible on this one.

Mandelbox Zoom from hömpörgő on Vimeo.

Revisting some old friends

I recently re-licensed a couple of my fractals for elementary school math books and decided it was time to revisit the old fractal program and see if I could come up with some new designs. I also discovered a neat batch rendering window which I hadnt noticed before which really helps me to crank out variations. Feel free to contact me if you would like to license any of these images.

 

Fire Fractal

Spikey Fractal

fractal spiral

infinite jewels fractals

striped spiral fractal

As a long fan of Ernst Haeckels work Kunstformen Der Natur – the forms of nature it was a real treat to find this Michael Hansmeyers platonic project this morning. Conceived in Processing, inspired by Ernst Haeckel, this project explores 3 dimensional subdivisions of topographic models recursively applying the subdivision process to a source form, which is restricted to one of the five platonic solids. A mouthful for you non-nerds I’m sure but the results speak for themselves.

Image Index | Project Index | Extra Images in an Animation

process3c

process2f

process4hprocess5c

Next Level Escher

Being the nerdy geeky sort I had a special fondness while growing up for the paradoxical nature, mathematical precision and infinite recursions present in MC Eschers lithographs, but recently I’ve grown to have an even more profound respect and admiration for his work from a fractal/loop perspective.

It started off with reading the article The Mathematical Structure of Escher’s Print Gallery by B. de Smit and H. W. Lenstra Jr. in which they analyze the print gallery as you see above where a young man is standing in a gallery looking at a painting that against all logic warps out and encompasses the gallery in which he is standing. There is a white spot in the center where the drawing stops. Why? Was it too complex to draw? Is there a mathematical paradox hinted at? The analysis of the original paper is pretty heady for those without a mathematics background and I hope the mathematicians don’t think of this article as blasphemy because of my attempt to make it simpler to grasp with my flimsy grasp on math. For those who have a solid grasp on math please don’t read my page (you’ll make me feel ashamed) and read the actual article instead.

The effect known as the Droste effect is named after a well known dutch chocolate in which the graphic on the box depicts a lady standing holding a tray with a box of Droste chocolate depicting a lady standing holding a tray with Droste chocolate… ad infinitum. This is known as recursion but how does this apply to Eschers piece?

The Print Gallery is actually a loop which contains a smaller version of itself, you start with a man in an art gallery looking at a print. This print hes looking at happens to contain the gallery in which he is standing but only 256 times smaller. Below is a video illustrating this part.

 

Why 256 times smaller?  First some math, but dont worry, it’s simple: 2 to the power of 8 = 256 (2x2x2x2x2x2x2x2=256) This means if you zoom in at a factor of 2X – 8 times to a certain point in to the image where there is a copy of the image 256 times smaller you will eventually end up back where you started. A seamless loop. As quoted from Bruno Ernst’s book  The Magic Mirror of MC Escher Escher started "from the idea that it must…be possible to make an annular (ringlike) bulge," "a cyclic expansion…without beginning or end." The realization of this idea caused him "some almighty headaches." At first, he "tried to put his idea into practice using straight lines [left], but then he intuitively adopted the curved lines shown in the image on the right. In this way the original small squares could better retain their square appearance."

  

What does this mean in laymans terms? Basically it means the small picture seamless warps into the larger one by using a grid to transform the image. Even cooler is that it creates a never ending seamless loop since it’s done with a zoom that ends up back where it started if you zoom in 256 times.

Below is one of the four original  studies for The Print shop.

Below are 8 images each one zooming into the previous picture by a factor of 2 (2x2x2x2x2x2x2x2=256) The rough areas are the white spot in the center of the drawing. By the time you get to the end you can see you are back where you started (the next zoom level is #1)

B. de Smit and H. W. Lenstra Jr of Leiden University in the Netherlands together with a team applied a 4 step process of reverse engineering this piece of Escher and reconstructing it

1) Decontrstruct – apply the reverse of the transformations to arrive at the original unaltered image.

2) Redraw the image

3) Recolor the image and apply the first logarithmic tiling transformation

4) apply the final transformation and the center is filled in!

While this is really cool and all I think what is the most amazing part of this is that different parameters can be used to create a number of variations of pieces escher might have come up with had he used different values. This was done by using a different transformation grid in the final step

By doing this the team created a large number of seamlessly looping zooming animations of what eschers print shop might have been – This page has all of them The animation below is the opposite values of what escher chose so it is in essence the yin to the print shops yang.

 One of my favorite visual mathematicians, Jos Leys has an amazing page explaining how to create the droste effect and use it in practical applications which will come in handy for those of us nerdy enough to learn how to put it into processing and code.

This page will be updated as I do more research on the subject.

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3D Animation Study


HD 3D Fractal Animation Study from Kris Northern on Vimeo.

 

This is a test render for a much longer piece Im working on which is currently rendering. I decided that I liked the end of this video the most and sought to replicate that sort of tweakery – So far the new animation is at frame 480 of 3200! It’s going to take a little while =)

Exploring Sierpinski’s Triangle

The Sierpiński triangle, also called the Sierpiński gasket or the Sierpiński Sieve, is a fractal named after Wacław Sierpiński who described it in 1915. Originally constructed as a curve, this is one of the basic examples of self-similar sets, i.e. it is a mathematically generated pattern that can be reproducible at any magnification or reduction. Given my curiousity for such sets I decided to use my 3D IFS modeling tool of choice: Xenodream, to explore a little deeper and see what I could find.

Iterations of sierpinskis triangle

The basic triangle is usually constructed in the 2D plane with these simple steps 1) Start with any triangle in a plane 2) Shrink the triangle to ½ height and ½ width, make three copies, and position the three shrunken triangles so that each triangle touches the two other triangles at a corner 3)Repeat step 2 with each of the smaller triangles infinitely (or until you run out of resolution) The next highest order is to do the same thing in the 3d plane. Xenodream ships with several starter .xep files (xenodream extension) which I used as starting points for these images I started by making this movie which is a seamless loop of a 360 degree rotation of the 3D Sierpinski triangle with some lighting effects. the movie is licensced under creative commons – please credit this page if you use it elsewhere

I started exploring this model from different perspectives and when viewed from the top down (as if it were sitting on a table and you were hovering right above it) I saw some very familiar geometries. The first thing I noticed is that it appeared isometric  (equality in dimensions or measurements) in this case – all the angles were the same (60 degrees) which is not a very large surprise considering the model is composed of equilateral triangles.

Sierpinski triangle viewed from the top

Sierpinski’s Triangle (all aspects equilateral) as viewed from above looking straight down. The center is the point.

I also noticed that the geometry was in Octaves and since the hexagons were so prevalent in the structure I started wondering if perhaps the Flower of Life could be mapped on top of this. Without getting too detailed the Flower of Life is a simple rule of repeating geometry by which all five of the platonic solids can be extrapolated – it is the basis on which all of sacred geometry is formed.  The image below shows the standard method of creating the seed of life – which if you continue adding circles after this to any point where a line intersects you end up with the Flower of life.

flower of life stages

Fruit of life

 

From the Flower of Life you can extrapolate the "Fruit of Life" which if you connect all the center points of these circles together you end up with Metatrons cube – of which you can selectively show lines to create all 5 of the platonic solids as illustrated below. What i find most fascinating is that this geometry is not only fractal but can grow in Octaves. Which makes it very useful for practical applications such as the creation of art work or creating structures that can scale infinitely which is useful for such things as the sand circles we used to do.

So, with that explained, I was curious if Sierpinski’s Triangle followed the same geometry as Metatrons cube. I had some initial trouble with the starter files as the placement of the triangles wasn’t exact, however after some very gracious work from Garth (creator of Xenodream) everything lined up and I was able to lay a Flower of Life grid on top of the render of Sierpinski’s Triangle as seen from above.

You can download this xep file for Xenodream here

As you can see below the teal lines  show that the geometry is also isometric which is also very handy for creative applications. Since the flower of life maps absolutely perfectly that means Metatrons Cube also will and anything derived from that as well. See the images below for some examples.

Sierpinski's Flower of life

The Flower of life and Isometric grid laid on top of Sierpinski’s Triangle as seen from the top (click to see full res)

 

Sierpinski <3's Metatron

Metatrons Cube Overlaid on Sierpinski’s Triangle as viewed from above (click for full res)

 

Sierpinskis Octohedron

Octohedral lines extrapolated from Metatrons cube overlaid on top of Sierpinski’s Triangle

What does this mean? Is it anything more than intellectual masturbation? I personally dont know, yet none the less I feel like I just bumped my nerdiness up a notch and will be able to lose people the moment I open my mouth. Most Superheros get some great superpower… I got stuck with the ability to make peoples eyes glaze over and look to the middle distance when I start talking. Oh well. So if you made it this far and still want more: Be sure to check out some illustrations I made (poorly and not 100% precise due to my lack of skill in Illustrator when they were made several years ago) of Octaves of the platonic solids as derived from Metatrons Cube http://www.phidelity.com/photos/v/Artwork/Sacred+Geometry/Octaves/ Or the article I wrote about Pythagoras Tree Dont Forget the Sand Circles derived from some basic Sacred Geometry I also asked what would Sierpinski’s Triangle look like if it had a square base rather than a triangular one and came up with this image which is viewed from the top. That’s all for this article! thanks for dropping by and be sure to leave your comments and thoughts!

Younger Brother / Shpongle poster

So it seems everyone in Boulder is rightfully getting psyched for the Younger Brother show on May 9th.

When I was last in Boulder, we hit upon the idea of doing a custom piece of art for a Conscious Alliance poster for this show. Most of you probably already know how great Conscious Alliance is, but for those of you who aren’t in the know yet, this is how it works: you bring 10 cans of food to the show and in exchange you’ll get a limited edition poster (created by yours truly), this food then goes to where its needed most. I think its brilliant, personally and I’m honored to put my energy towards the cause.

So I wanted to write a little bit about the design I ended up creating, but I should warn you now, I don’t end up showing the full design and I kinda start nerding out, but that’s just how I roll.

My main area of interest lies in 3D fractals – These are 3D objects that can rotate in 3 dimensional space. The banner at the top of this site was made of a collage of different 3d fractals I created with a PC based program called Xenodream. While most standard fractals are 2D, even the ones you can zoom into – the addition of a 3d dimension allows for much more expression and control in creating images. I started off by remixing the 3d fractal I used for the Younger Brother flyer and spent a few days rendering it (it takes a looooong time to render the size I need to do full resolution 18×20″ print) – This is a section of the fractal – Click to see more detail

Once that was done I started collaging it all together, I hit up all my friends to see who had a copy of the first Shpongle album to scan it for me so i could integrate the shpongle mask into the design, I am a bit shamed to admit I just recently sold my copy of it on vinyl to fund my gear addiction. From there it was some simple design work to make the whole design cohesive. Below you’ll see some sections from the design and some links. That’s all from me – If you want more you can always check out my site.

xoxoxo kris phidelity

more 3d fractals

animated 3d fractal video of Rena Jones – Open Me Slowly

 

 

 

Pythagoras Tree

So it all started with a challenge that was posted on the Xenodream pix list to work with the Pythagorean Tree. As illustrated below, The Pythagoras tree is a plane fractal constructed from squares. It is named after Pythagoras because each triple of touching squares encloses a right triangle, in a configuration traditionally used to depict the Pythagorean theorem. Garth, the creator of Xenodream had posted the initial .xep file with the simple repeating squares. From the left each image is an iteration of this function growing exponentially more complex. The top is a 45,45,90 triangle and the bottom is a 30, 60, 90 triangle resulting in a lopsided tree.

Pythagorean Tree

Since squares are 2 dimensional I decided to add a third dimension by making the squares into cubes and exploring the geometry in 3 dimensional space. The tree itself was interesting but since I’m reading Samuel Colmans “Harmonic Proportion and Form” I decided to test some of the ideas he’s pushing in the book; I decided to play with angles derived from the geometries of platonic solids and lo and behold I got a Dragon Curve. The cubes were rendered 95% transparent so you can see the geometry inside of it.
Pythagoras Fractal

(above) Pythagoras Dragon Curve .xep
(below) Same image with outlines instead of cubes and not viewed from an angle. .xep

Pythagoras Dragon Curve

 

I continued to play with the angles of rotation I found an interesting setting: if you rotate the left square 135 degrees and the right 225 degrees you end up with perfectly nestled fractal geometry of the Pythagorean Triangle, which ends up extending and filling a space proportionate to the original triangle. The squares themselves are subdivided by four of the triangles illustrating their geometric relation. I suspect the white lines are due to the manner we are drawing lines in Xenodream and that perhaps the geometry would fit together perfectly if I were to model it more precisely, which I will probably do with Processing at some later date. Click the image to explore the 4000+ pixel render of this image!

Pythagoras Triangle Fractal Geometry

Pythagoras Triangle Fractal (above) and close up of detail (below) .xep

 

Below are a few other images I made playing around with this shape

geomeTree

Below is a side by side comparison of the by now familiar tree and below that the same geometry with a little variation; instead of a square I created a shape that was very very vaguely trunk like. I apologise for the mid ’90s 3D quality of this image I wasn’t setting out to create a realistic tree, Just to see how convincing the branching patterns were. And for the little amount of work that went into it I’m impressed.

very 90’s looking 3D tree .xep

While researching for this article I found an artist named Koos Verhoeff who has made a bronze sculpture using this base geometry.

Bronze Sculpture of pythagoras Tree

All the xep files which are for Xenodream are released under a WTFPL public license and you can do whatever you want with them.
That’s it for now! If you’ve made it to the bottom of this page you have a special sort of geekiness that would probably enjoy my RSS feed of random nerdy and geeky things like this. Look near the top right

Fractal Overdose

Ive been playing with a slew of fractal programs generating images for projects and testing scripts and what not below is a random assortment of images I came up with.

 

Mandelbrot Fractal

Expansive Flowering – made with Fractal Explorer

 

Mandelbrot Deep Dive

Mandelbrot Deep Dive made with Ultra Fractal

 

Mirror Shards

Mirror Shards created with Apophysis

 

Evil Fractal

evils made with Xenodream

 

Flourite

Flourite fractal made with Apophysis

 

Bipolar Fractal Flame

Bipolar Fractal Flame made with Apophysis

 

Thimbles Flame

Fractal Flame Made in Apophysis