fractal forms


Fractal Cities - A Geometry of Form and Function

This book presents an initial attempt to apply fractal geometry to cities. In fact, we go beyond this and argue that cities are fractal in form, and that much of our pre-existing urban theory is a theory of the fractal city.

In terms of theory, we show here that the architect’s physical determinism concerning the city can be captured and elaborated in terms of fractals while the geographer’s concern for the economic theory of location is entirely consistent with the use of fractal ideas.

© Michael Batty ©Paul Longley

Meshes of starlight like woven cobwebs hold fire within their endless loops of patterns, multiplying fractals, breaking apart, forming planets and asteroids from spheres of flames that are constantly searching for a place for themselves in the darkening spilled void.
Geometric patterns of realities tend to infinity in parallel universes where every word echoes with abstract tastes, ricocheting off meteors endlessly, never finding a place to fit in the vacuum that bends with the weight of infinite geometric progressions looping themselves over and over and over until there’s nothing left but curving cobwebs of space and time, expanding with every breath and crackle of starlight.
And here we are, stuck in this reality, that is both harmony and chaos, burning and freezing.
And here we are, stuck in this sphere of flame that is learning to find a place for itself before it all burns out again.
—  Tamarind Fall; Infinite geometric progression.
NaPoWriMo day 6.
Constellations in the Sunlight (Adlock)

The salty ocean breeze blew lazily against his cheek and the waves lapped gently over the shoreline, not quite reaching his feet. It was just the two of them. The others were gone.

His mother was sharing drinks with her aunt somewhere on the other side of the beach, the former sitting under the shade of a large umbrella, and the latter attended to by her lover, the Spanish poet. 

His mother was no doubt breathing a sigh of relief for the reprieve from the responsibility of caring for her eccentric teenage son, and for the fact that he was spending time outdoors with another young person his own age, rather than cooped up inside the house, cooking up his experiments. And her aunt simply could not be bothered to tend to her niece’s needs when her own superficial needs were, to her mind, much more pressing.

He lifted his face up toward the sun and squinted against the sunlight creating brilliant fractals in his vision. He was about to close his eyelids against it, but a shadow moved into his line of sight and slowly took form as she hovered over him.

He could see the unique shape of her begin take form within his vision – the fall of her unruly dark hair, the sharp corners of her thin shoulders, the supple lines of her youthful arms.

He could not see her face, but that was the beauty of it. 

The darkness formed by shadow against sunlight hid her face from his, creating a vacuum, an inverted constellation, through which she gathered all light within her and let none escape. The fractals of sunlight formed strange patterns around her, each vying to be a star in her complex astronomical cortege.

He lifted his finger and traced her outline, his own personal constellation – a phenomenon that appeared only in the summer sunlight on a beach in the south of France. 

“No stars in London,” he mumbled, barely aware of what he was saying. He could smell the scent of the perfume her aunt forbade her to wear clinging to her skin, and he inhaled deeply. “Can’t see them there.”

He heard her laugh softly as she took his finger and guided it to her lips, still hidden by ochre shadows. She pressed a kiss to his fingertip, and he felt her smile against it.

“Well, perhaps we should watch them again tonight.”


By SorrowsFlower

Excerpt from this Adlock teenlock fic I’m planning, tentatively called “Beautiful Children”, based loosely on the Chanel short film, Tale of a Fairy by Karl Lagerfeld.

Thanks to @randombiochemist for giving me the constellation idea!

A Length of Hope

By Roshani Chokshi

The circus was underwater. Crowds would set up tents beside placid lakes weaving between mountain hollows and rabid rivers foaming white at the mouth. They would wait for days, hearts in their throat as they waited for that diamond-studded fin to slice through the water. A herald. Or was it a warning? Angelique could never be sure. All she knew was that it meant the circus was going to start.

Once the fin appeared, the scaffolding rolled to life. The water would rise, pouring through the air and forming steeples and staircases, amphitheaters carved like crescent moons and mirrors that broke the sunlight and scattered them into spotlights for the performers. It was always like this when the underwater circus visited. They had no name. They took no payment. Sometimes a person went missing — usually a teenage boy — but no one paid the missing boys any mind. Magic came at a cost, and most people were willing to pay with someone else’s grief.

Angelique remembered the first time she saw the circus perform. She sat at the very edge of her seat, leaning so far out that her body ached the next day from straining. There was so much to watch — the rusalka ballet, the mermaids’ synchronized swimming, the leviathan’s magic show.

When the rusalkas emerged from the water with their glassy ankles beribboned with seaweed and strung with lumpy river water pearls, Angelique would forget to breathe. Her feet itched to dance like them…skimming across the water’s surface like a thread of light, the silver hair streaming behind her, spangling the air with frost. When the mermaids began their synchronized swimming, Angelique would forget her name. She wanted to dive through the water with them, splashing a gemstone tail and forming fractals and kaleidoscopes with her body alone. And when the leviathan’s magic show started to conjure glass horses chasing each other across the lake, Angelique would forget her life altogether.

But her favorite part of the show was the Sea King. She wasn’t sure if he was actually a king of the sea, but he could have been. He was short in the way a mountain in the distance looked almost edible. Human-sized. There was something restrained and folded about him, a self-conscious fidgeting as if he couldn’t fit in his skin.

The Sea King ran the circus, pointing and directing with little more than a nod and a blink. He wore a crown, or something like that. Small lights danced around the fall of his teal hair, sometimes clinging to the sharp gold of his cheekbones that were lightly flecked with scales. Sometimes the lights danced in his strange eyes — all-black pools holding all the enigma and horror of an open ocean.

Angelique loved him. She loved him the way people love rain or a full moon. A distant love, with no desire to capture and possess. And yet it was precious. Night after night and day after day, she would return to the memory of him, smoothing it over and over in her heart as if she were wearing down a pebble. Night after night and day after day, that love filled her, turning her heart incandescent.

The circus never visited the same river or pond or lake twice. But Angelique always knew where they would be. She would catch a snippet of the next venue in that haze of heavy comfort that always summons sleep. She would replay the image in her dreams — clinging, clinging — until she could wake up and search for the place in whatever public library lay closest to her newest home. Then again, a broken mattress and jackets fished out of dumpsters for blankets was hardly a home. But it was warm. Every time the dream found her, she would spend a week scraping together as much money as she could. The circus never started without her.

Maybe it was waiting for her, just as she was waiting for it.

The performers of the underwater circus never spoke. When they finished their routine, they would stand on the surface and gaze out at the cheering audience. They would bow, and then, they would start to shimmer. Veins of quartz and fire would thread across their bodies, turning them into glass dolls on the verge of shattering. But right before they broke, they would always disappear.

Angelique wanted to be one of them. Her want felt so large that she was a shadow trailing after it. She never thought they would notice. Why would they? They were beings of light and water, things cobbled out of myth and sleep.

But one day they did.

She never saw the Sea King move toward her. One moment he was standing at the edge of the lake, his blue skin polished as a sapphire, fresh daylight streaming through his body. The next he was at her side. He had a handful of rope in his hand, and when he traced her palm, her lungs tightened. As if she were underwater and could draw no breath.

He didn’t speak, but the rope in his hand was an invitation given shape. Something twisted in the threads — a chance. When she closed her eyes, she could feel the possibility pressing palms to her eyelids. A life forever beneath the water. Gills blooming on her neck. Cold water rushing to fill her mouth and braid her hair. She could swim through their secrets, know what underwater realm inspired the watery spires and crescent-shaped amphitheaters. The Sea King was giving her the chance of a secret.

That is the problem with the taunt of a secret. Once you know they exist, they grow fat in your heart. They set down their roots and sprawl through your thoughts until you can hardly speak around wanting to know the thing you do not. The Sea King knew this.

“It is not so hard,” he said slowly, “to bind your body to a piece of hope. You’ll need a new one anyway.”

 Angelique is taken with the piece of rope. She is taken with the Sea King’s eyes latched onto hers — dark as an eclipse and just as overpowering. This secret is an eclipse too. A celestial thing slipping before a stale light and casting her thoughts into darkness. She is so taken that she does not see the lean hunger straining his eyes. She does not see how his lips are chapped and flaking, as if he had licked them too often. She does not see how when he smiles, his teeth draw blood.

When the rusalkas dance, she watches. Slow bubbles tease their way out of her lips.

When the mermaids swim, she watches. Blood pounds in her ears.

She thinks of the Sea King’s promise. Around her, the watery light shows shapes of bodies on the lake floor. Young men, mostly. What is left of their hands is still bound. She does not know if this is the body they left behind for something sleek and new. Or whether this was it.

When the leviathan performs, she watches.

The lake is still and humming. The water a symphony of dreamsweet things, plucking at her eyelids, forcing them to close. Soon, the Sea King will come and get her. He will free her of these mortal bounds and pour her soul into something new. As soon as she wakes, he will be there…

        As soon as she wakes…

        As soon as she wakes…

        As soon as she wakes…

Roshani Chokshi is the author of THE STAR-TOUCHED QUEEN, a Hades and Persephone reimagining with Hindu mythology. Her next book, A CROWN OF WISHES, comes out March 2017. 

Imagine the Universe forming a fractal counterpart. A mini counterpart that exactly resembles it. Imagine this cosmic universal energy to be a glowing cloud of hyperdimensional gas, swirling and dancing with multi-hued colors. Now around this hyperdimensional cloud, wrap and assemble a physical body, bone by bone, muscle by muscle, until it completely encases this cosmic dancing cloud within it. Now open your eyes and look in the mirror.

In the haze of your fixation, I’d forgotten what gravity tastes like.
My arms spread wide, fingers turning white, the cold wind gripping my skin and nothing but the ground way under my feet.
There was no snowfall. Just the fractals forming a crystalline web, trapping me mid-air, turning the whole world into a massive mesh of frosted glass I could not see through.
One crack and it all fell apart.
One word and the web of lies you’d been spinning all around me collapsed and I fell to the ground with nothing but panic gripped tight within my fists, held against my head, hoping I don’t break when I crash.
Betrayal tastes like the metallic taste of blood on your tongue. Like bones starting to crack along the edges from holding on too tightly for too long. Like the numbness in my feet after the fall.
Realisation cut me deeper than the shards of glass digging into my skin and every step I took was a desperate attempt to escape.

It was difficult to walk back home.
I spent months trying to carefully pick the shards of glass out of my head so they wouldn’t keep piercing me on the days I couldn’t even remember my own name.
I spent days sweeping all the memories scattered around my front porch, afraid of getting hurt again.
Now, I see you look back at me. With your frosty smile and crystal blue eyes that briefly reflect regret before they turn icy.
You tell me you’ll heal all the bruises lining my skin or you call up on the nights you find my bedroom lights still on, the soft light spilling on the streets, maybe trying to see if I still hurt where you touch my sores.
But I’m too numb, having vowed never to forget how the gravel still chafes against my bandaged feet.
—  Tamarind Fall; Writing prompt: You betrayed me and now you look back, either with regret or just to get a reaction but I’ve moved on.

The main (central) sets of 7 chakras form a fractal system, with an infinite set of similar ones (both upwards and downwards), but less and less significant the further away they are from the main set. The chakras are larger in extent than I’ve illustrated here when in good condition, and each chakra has components one of which spins clockwise, one of which spins anti-clockwise. The chakras are in part a form of energy-transformer, supplying different frequencies of energies to the meridian-system and other body-systems. Other functions include communication and awareness at their range of frequencies.

The main set of chakras (i.e. chakras for meat-body) has 7 chakras. The lowest is red and is about physical survival, material existence etc. The second up is orange, and is about sex and reproduction. The third is yellow and is about power. The 4th up is green and is about connections with other things. The 5th is turquoise, and is about communication. The 6th is blue and is about awareness, and the top one is indigo and is about connection with higher things (“spirituality” etc.).

As on the illustration, above the main set is another set of 7, which are the chakras for the Etheric-double. Some belief systems mistakenly see the first one of those as an 8th one of the main system. Below the main chakra set is another set for the Astral-body. These sets other than the main set are not really the colors I’ve illustrated (and the main ones only relate harmonically to the illustrated colors, as they are not in the Physical-universe [1] Playing-field), but are something like related harmonics of those colors.

Reiki and Hatha-Yoga work on the chakras, but traditionally only on the main set. 

In addition to the sets of 7 chakras which are on the vertical energy-line (which is infinite both upwards and downwards and is where some of the energy transformed by the chakras mainly comes from), there are many other chakras (some of which are illustrated here). The main additional ones are on the joints, and these are worked with in Reiki, plus there are series’ of chakras extending out from each limb (as illustrated) The knee chakras (and the chakras in the series below each knee-chakra) are particularly important for the power of the Kundalini.

The main sets of chakras have input (Source) from behind the body, and output (Ground) towards the front (in addition to their connection to the infinite energy-line). This flow can be cleaned somewhat by temporarily reversing it for a few seconds, then putting it back to normal.

The additional chakra-systems outside the 3 ones labeled here, have each set connecting somewhat to a chakra in the main sets.

Some versions of Hatha Yoga suggest a particular symbol for each chakra. While these symbols can be of use initially to help clean and focus the energy of each chakra, one should not use them beyond that point, as they are too narrowly focussed, and not precisely correct, for the perfect opperation of the chakras.

There is at least one additional Chakra with different properties to those illustrated, so could be considered either as a Chakra, or as a separate type of body-system (although some Eastern traditions use the word “chakra” for any energy-formation, such as the Brains etc.). This is the KalaChakra (not illustrated), which has its origin behind and slightly below the Solar-plexus (although its shape can extend upwards into wing-shaped formations, illustrated (inaccurately) in many traditions). None of the “KalaChakra” traditions on earth currently are working cleanly or directly with the real KalaChakra.

These are well known and used by the acupuncture tradition, and relate to different organs, both organs of the physical-body and “organs” of the energy-systems. However, they can also relate to other energy-bodies (for example one meridian which is near the surface of the skin on the side of the neck connects to the 3rd Soul-level). Also, there are actually more meridians outside the body (not illustrated) than inside it, and each meridian junction has a connection to Source/Ground.


Snow is a natural phenomenon. It  consists of the precipitation of small ice crystals with fractal geometrical forms when high concentrations of water vapor accumulates in the atmosphere at temperatures under  0°C. It is only then when snow happens!!!

But one of the most curious and incredible things about this white coat, which you generally see in your back yard during the christmas eve, are the beautiful and perfectly geometrical shapes of the snow crystals. If you take a detailed look to one of this pictures, you are going to notice that every single snow flake has faultless hexagonal geometry, ( each one with six arms) with no exception. Isn’t it amazing? These are master pieces of art of the most perfectionist artist: nature!!!

All the photographs in this post were taken by Russian photographer  Alexey Kljatov, we do not own this material.

For more interesting and fun facts , follow musicat9419


Unfinished eye color reference drawing, doodle I drew to Muse’s “Big Freeze” (I adore that sooong) and a really stupid sketch that shows how little I know anatomy hahahaha.

I’m still really fond of the markings Ranul and Ralos used to have on their chest in their original designs back in 2008, I’m trying to keep it for nostalgia reasons!
Also forgot to add in that Ranul’s thin golden vein-like markings are created from the Iolaria virus. It distributes either in fan/crystal-like organic forms or in fractal/geometric-like artificial forms. Still working out the details! @u@

Some time has passed since I posted about a mathematician and I am incredibly happy to come back to this project with such a wonderful person.

Benoit B. Mandelbrot, born on 20th November 1924, was a Polish-born, French and American scientist-mathematician. He has been most widely recognized and honored for his discoveries in the field of fractal geometry.

In 1975, Mandelbrot coined the term fractal to describe these structures and first published his ideas, and later translated, “Fractals: Form, Chance and Dimension”. According to mathematics scientist Stephen Wolfram, the book was a “breakthrough” for Mandelbrot, who until then would typically “apply fairly straightforward mathematics … to areas that had barely seen the light of serious mathematics before.” Wolfram adds that as a result of this new research, he was no longer a “wandering scientist”, and later called him “the father of fractals”:

<<Mandelbrot ended up doing a great piece of science and identifying a much stronger and more fundamental idea—put simply, that there are some geometric shapes, which he called “fractals”, that are equally “rough” at all scales. No matter how close you look, they never get simpler, much as the section of a rocky coastline you can see at your feet looks just as jagged as the stretch you can see from space.>>

Riemann zeta function: correlations between the properties of the local maxima of the Riemann-Siegel Z function and the properties of some fractal features

In my previous post I provided some initial evidence of a remarkable correlation between the number, position and relative size of the major butterflies of the fractal formed from iteration of the Riemann zeta function, and the number, position and relative amplitude of the maxima of the Riemann-Siegel Z function.

I promised to provide further evidence for the correlation by showing a map of a region of the fractal at a height of around 2447i where there is a particularly large butterfly sandwiched between two particularly tiny butterflies, a configuration reflected in the corresponding section of the graph of the Riemann-Siegel Z function.

Here is the image:


Fascinated by the intricate patterns formed by fractals, basically math processes that repeat incessantly in an ongoing feedback loop, UK physicist-turned-web developer Tom Beddard makes impossibly elaborate complexes that look like they belong in the gritty cities of a dystopian fantasy. Basically what he does is write and run programs on his computer that spit out patterns—"the best outcomes are often the least expected!“ he writes—that he in turn massages (by way of shadowing and the like) into looking like faceted steel-and-concrete architecture.