Latööcarfian Explorer (Scratch ed.)

I have made a Scratch project that generates Latööcarfian images from chaotic attractors as described in Clifford A. Pickover’s Chaos in Wonderland: Visual Adventures in a Fractal World (St. Martin’s, 1994). It is based on the BASIC listing on page 268 of the book. Link to project page where you can view the complete instructions.


These are the same kinds of images I wrote about here. I have used them as the magical names of entities I have evoked.

Generative Sigils

Related articles: TAD2011.06 Lorenz Attractor | Processing + EPOC via OSC

In his book, Chaos in Wonderland: Visual Adventures in a Fractal World, Clifford Pickover describes methods for generating beautiful, complex images from certain chaotic equations. In the context of the book’s narrative, these images are the dreams of a species of inorganic, computer-like entities called the Latööcarfians — the “dream-weavers of Ganymede.” Here I consider using these images as algorithmically generated magical sigils (cf., generative art).

The images are generated by recursively plotting:

xt + 1 = sin(ytb) + c sin(xtb)
yt + 1 = sin(xta) + d sin(yta)

(There are variant equations that produce “mutations” — see “Appendix A: Mutations of Equations”, pp. 209–210.) Here is a sketch that will draw the following image in Processing:

/** Generative Sigil 1
 * Joshua Madara,
 * Based on code on pg. 26 of _Chaos in Wonderland_
 * by Clifford A. Pickover
 * Good ranges for a, b, c, and d:
 * (-3 < a, b < 3)
 * (0.5 < c, d < 1.5)

float a = 1.5641136;
float b = 2.7102947;
float c = 0.9680385;
float d = 0.995141;
float x, y = 0.1;
int counter = 0;
int iterations = 500000;

void setup() {
  size(700,700,P2D); // remove P2D for Processing v2.0
  background(0); // black
  stroke(255,255,255,90); // white, semi-transparent

void draw() {
  translate(width/2, height/2); // draw from center of window
  float xNew = sin(y*b) + c * (sin(x*b));
  float yNew = sin(x*a) + d * (sin(y*a));
  x = xNew; y = yNew;
  point(x*100, y*100);
  if(counter >= iterations) {

Generative Sigil 1

I hypothesize that the key to using such images successfully as magical sigils is to assign non-trivial values to the inputs, a, b, c, and d. E.g., one could randomly generate the values at an auspicious moment, or acquire values from some act or object, and map those to the optimal ranges for the algorithm’s inputs. The magician could wear the Emotiv EPOC during a magical ritual and at the ritual’s apex a Processing sketch could map data from the EPOC to the a, b, c, and d values for generating the image. The images could subsequently be used for divination or evocation.

N.b., even while keeping the input values within optimal ranges, not all sets of values produce interesting images. Here is a Processing function to calculate the set’s Lyapunov exponent (based on the code on p. 62 of Chaos in Wonderland) — values >= 0.5 tend to be more interesting:

float calcLyapunovExponent(float a, float b, float c, float d) {
  float Lsum = 0;
  float n = 0;
  float x = 0.1;
  float y = 0.1;
  float  xe = x + 0.000001;
  float ye = y;
  float xx, yy, xsave, ysave, dLx, dLy, dL2, df, rs, L = 0;
  float bigNumber = 2139095039; /* Pickover's algorithm calls 
     for a long int (1000000000000) here, but I often get NaN returned 
     when using it in Processing, and I have found that using a 
     large float returns a value close enough to Pickover's to 
     be useful. */
  for(int i=0; i<10000000; i++) {
    xx = sin(y*b) + c*sin(x*b); yy = sin(x*a) + d*sin(y*a);
    xsave = xx; ysave = yy; x = xe; y = ye; n++;
    xx = sin(y*b) + c*sin(x*b); yy = sin(x*a) + d*sin(y*a);
    dLx = xx - xsave; dLy = yy - ysave; dL2 = dLx*dLx + dLy*dLy;
    df = bigNumber*dL2; rs = 1/sqrt(df);
    xe = xsave + rs*(xx - xsave); ye = ysave + rs*(yy - ysave);
    xx = xsave; yy = ysave; Lsum = Lsum + log(df); L = 0.721347*Lsum/n;
    x = xx; y = yy;
  return L;

MagiCalc 2

I have created a new version of the Magical Probability Calculator. Edited (2015-12-29) to say you can view the code here, but it was written for Processing 1 and no longer works in 2 or 3 as-is, nor does the online Java applet work now. However, I have recently made a JavaScript version sans Processing.

MagiCalc 2 Screenshot

Changes in v0.2:

  • Updated m calculation to reflect changes made in Octavo.
  • Fixed NaN errors for various combos of p=1, m=1.
  • Added toggle for amplification/attenuation.
  • Added dynamic text displaying calculations.
  • Added graph to plot p_m.
  • Added percentage views for glsbm.

Enhancement ideas:

  • Add a toggle and slider for countermagic. (I had intended to do that for this release, but the logic to select ATT when e.g. the countermagic (M_C or Mcontra) exceeds M, was blowing up controlP5, and I did not find an elegant solution before the time I wanted to publish the new version.)


Electronomicon Glow

Here are some images and details about my art exhibit at this year’s Esoteric Book Conference. The project demonstrates some possibilities for computational magic and telematic and automatic telesmata, and is intended to provoke discussion about the expression, automation, and trivialization of magic through technology. The basic concepts underlying the technology are based largely on several of Peter Carroll‘s ideas and include:

  • The probability of an event’s occurrence can be intentionally influenced through magic, and gnosis, link, subliminalization, and belief factor into such influence, which is called enchantment. [1]
  • Gnosis, subliminalization, and belief may be effectively transferred to or otherwise replaced by an artifact for automating magical agency thus requiring only an adequate link. [2]
  • The best magical link is a real-time line of sight to the target, which can be facilitated across occluded space by electronic vision. [3]
  • The material basis (telesma) of a magical artifact can physically interact with electronic systems relating to the artifact’s magical interactions with ætheric systems.
  • Color and sigil magic.

The project began with the Electronomicon and then grew into PsiBorg: Enchantment Amplification and Automation System, of which the Electronomicon became a component.

PsiBorg 1
(Looks like a science fair project, doesn’t it?)

Media, by altering the environment, evoke in us unique ratios of sense perceptions. The extension of any one sense alters the way we think and act — the way we perceive the world.

When these ratios change, men change. // Marshall McLuhan and Quentin Fiorem, The Medium Is the Massage: An Inventory of Effects

PsiBorg 1

…where man’s word goes, and where his power of perception goes, to that point his control and in a sense his physical existence is extended. To see and give commands to the whole world is almost the same as being everywhere. // Norbert Wiener, The Human Use of Human Beings: Cybernetics and Society

PsiBorg 1

…there is the possibility that we may progress towards making artifacts so remarkable that they utterly transcend our present ideas of what constitutes a ‘machine’, and of what a machine’s limitations must be. // Ramsey Dukes, Words Made Flesh: Virtual Reality, Humanity, and the Cosmos


The Electronomicon is a book of seven printed-circuit-board (PCB) talismans representing seven magical servitors corresponding to the seven non-octarine colors of Chaos magic [4]. The talismans’ electrical conductivity allows them to act as switches in electronic systems (either passing a current or not), and also allows for novel forms of ritually “charging” them.

The sigils were drawn by Dakota Crane, then etched and consecrated by myself at auspicious times. The names of the corresponding servitors were divined during a series of Ouranos Rites [5], and used to charge the talismans by transducing the intonations of the names to analogously variable voltages connected to the corresponding sigils. The servitors were evoked by Meta-Magick techniques [6] that were combined with the process for charging the talismans.

The talismans are connected to PsiBorg using the red (+) and black (-) clips.


Here is a sigil-machine that Dakota constructed from the Electronomicon sigils, “a fully-functioning psychotronic device made to facilitate interaction with the concept of Robomancy” (and here is his documentation for it):

Machines, Take Up Thy Schematics and Self-Construct!

(There are still prints available of Dakota’s sigil-machine. They are 8.5 × 11 inches, full-color, and come with a printed copy of the documentation on cotton paper. $10 each, including shipping. Email me if interested:

Roboticus Familiaris

Robotic Familiar (Roboticus Familiaris)

The robot supplies telepresence to the magician who can see and hear through its “eye” and “ear” — like a witch looking through the eyes of her familiar animal or spirit. The owl skulls represent the powers of sight and hearing (the resin replicas were created by Ann Koi—thanks, Ann!). The robot’s UV LED can light up otherwise invisible sigils and other occult markings. The arrow keys on the computer keyboard move the robot forward, backward, left, or right, and acquire a view of the target (I ended up favoring keyboard control over the NaviGlyph system I had developed earlier).

Roboticus Familiaris

Roboticus Familiaris represents the possibility of acquiring a line-of-sight magical link by mobile electronic vision. Although the robot demonstrated here is tethered to a computer and controlled by a keyboard, systems for wireless control and autonomous locomotion are well known and available. Telepresence robots are on the rise (see e.g. Anybots).

Please see this article for more about how the robot is driven by an Arduino microcontroller and motor shield.

Oculus Electricus

The Eye Electrical

Physical contact or line of sight to the target provides the best possible form of magical link as it minimises the imaginary temporal separation and it also provides the opportunity to visualise the target in real time. Some magicians refuse to work with anything else. […] Real time links across occluded space generally require the use of visualisation or telesmata although an additional real-time connection via an electronic communication network often helps. // Peter J. Carroll, The Octavo (emphasis added)

The Oculus Electricus is a digital video camera mounted on the robot, which represents a servitor’s-eye-view of the magician’s target, extending her vision through occluded space and augmenting it with colors and sigils of magic. The color scheme automatically changes to correspond to the selected Modus Magici (see below) — i.e. the device (or the magician through the device) “sees red” in war magic mode, orange in thought magic mode, blue in wealth magic mode, etc.

For the exhibit, I placed Roboticus Familiaris + Oculus Electricus in a half-covered box with a doll having a fluorescent green head. PsiBorg is programmed to “recognize” the doll’s head while in AutoMage mode (see below), thereby simulating facial recognition.

Oculus Electricus

The video signal could be streamed wirelessly or over the Internet, establishing a telematic link between the magician and her target, in addition to a sympathetic (magic) one.


PseudoHUD is a visual interface to the PsiBorg components, which feigns a heads-up display.



Based on Carroll’s equations of magic, this component calculates (m)agic and magically modified probability (pm) from the factors (g)nosis, (l)ink, (s)ubliminalization, (b)elief, and (natural) (p)robability. The values can be adjusted by using the mouse to move the slider controls. In AutoMage mode (see below), g, l, s, and b are automatically set. Two bar graphs show pm+ and pm− within the Oculus Electricus viewer; they respectively represent magical influence in favor of a desired event or opposed to an undesired one. As m increases, pm+ increases and pm− decreases. (See this article for more about the calculations.)


The magician can generate an animated, three-dimensional sigil layered over the video image from the Oculus Electricus by selecting up to sixteen alphabetical characters from the keyboard. Each Modus Magici (see below) has a unique array for generating sigils appropriate to it, formed from the name of the associated talisman’s servitor. (See this article for more about how the 3D sigils are generated.)


The magicikian will create a reusable multi-task servitor by such conjurations, for which the first equation’s factors take an optimum unitary value. Thus only the problem of a magickal link awaits solution in any particular spell of enchantment or divination at which the magickian directs the eidolon. // Peter J. Carroll, Psybermagick

When AutoMage is activated by pressing the F1 key on the keyboard, MagiCalc variables g, s, and b are automatically set to 1 (100%) when an Electronomicon talisman is electrically connected to PsiBorg, and l is set to 1 (100%) when a target is acquired by the Oculus Electricus (i.e. the green head of the doll is within the camera’s view). This behavior represent the automatic efficacy of a servitor or talisman.

Modus Magici

PsiBorg has seven magical modes corresponding to the seven talismans of the Electronomicon. The magician can switch between them by pressing the number keys on the keyboard: 1=Black/Death/Saturnine; 2=Blue/Wealth/Jovial; 3=Red/War/Martian; 4=Yellow/Ego/Solar; 5=Green/Love/Venusian; 6=Orange/Thought/Mercurial; 7=Purple/Sex/Lunar. In addition to a unique color scheme, each mode has its own sigil-generating array based on the name of the servitor associated with the talisman corresponding to the mode.

Open Sourcery

All of PsiBorg’s components were made with open-source resources including Processing, Arduino, GIMP, and Inkscape, and I expect to publish source code, bills of material, and rituals here on hyperRitual in the near future, in accordance with the principles of open-source magic.


Overall, I felt the project was well received. With so many moving parts, it was difficult to document how it was supposed to work without overwhelming my audience (which I suspect I did anyway). Things went better when I was able to personally demonstrate its operation, and I am very grateful for those conference volunteers who enthusiastically showed it while I attended to other matters. My biggest fan may have been a young girl who was not shy to interact with all of the pieces and who returned several times to do so. Two respectable publishers showed interest in publishing my robomancy book.


  1. Peter J. Carroll, Liber Kaos (Boston: Weiser Books, 1992) 40–51. The Octavo (Oxford: Mandrake, 2011) 108–112, 116–121.
  2. Peter J. Carroll, Psybermagick (London: BM Sorcery, 1995) 24–25.
  3. Peter J. Carroll, The Octavo (Oxford: Mandrake, 2011) 105–106.
  4. Peter J. Carroll, Liber Kaos (Boston: Weiser, 1992) 107–141.
  5. Peter J. Carroll, The Ouranos Rite (Tempe: New Falcon, 2003).
  6. Philip H. Farber, Meta-Magick: The Book of Atem (San Francisco: Weiser, 2008).

Magical Probability Calculator

Related articles: Psyleron REG-1 | Empirical Evidence of the Efficacy of Sex Magic?

Here is a Processing sketch that calculates magical probability per Peter Carroll’s first three equations of magic from Liber Kaos [1]:

  1. M = GL(1 − A)(1 − R)
  2. Pm = P + (1 − P) × M1/P
  3. Pm = P − P × M1/(1 − P)

Magical Probability Calculator

Edited (2016-01-13) to remove link to Java applet version since it no longer works predictably. Here is a somewhat improved version, also made with Processing, and here is a JavaScript version.


Use the blue sliders to manipulate the variables. The green and red bars show how natural (P)robability is adjusted by the amount of (M)agical power — (G)nosis and (L)ink increase magical power, (A)wareness and (R)esistance decrease it. The green bar shows magical influence on manifesting a desired outcome (→ 1.0); the red bar shows magical influence on preventing an undesired outcome (→ 0.0). Both bars become brighter as magical power increases.

Please use the comments section below to leave feedback or ask questions, or use the contact form.

Notes & References

  1. 1. Peter J. Carroll, Liber Kaos (Boston: Weiser Books, 1992) 41–51. In The Octavo, Carroll changed the formula for M to GLSB, i.e. Gnosis, Link, Subliminialization, and Belief, each from 0 to 1, and all multiplied together.