Circular/Spherical Infinity

?: David Eppstein told me "You can get something resembling your fractal using an inversion process like the ones used in the above pages, starting with three mutually tangent circles, but unlike yours the circles at each level of the recursion wouldn't all stay the same size as each other."

?: Lucian Wiznick has finished his screen saver (1.2MB) based on this image. It does part of what I'd like to do, and even some things I hadn't thought of (like perpendicular rotation). Now if only I can get full control over the individual inner circle rotation timing and maybe even in true 3D (spheres vs circles), that'd be cool... Anyway, thanks Lucian!

~1999/12: Created webpage and counter.

2000/3/21: ?
2001/12/7: Added last 2 paragraphs (now "logic" and "absolute relativity" applications).

2002

2/27: Found a recursive circle drawing made with DrawScript (a RiscOS program). Created spherical infinity Maya model.

3/16: Asked in some fractal newsgroups for fractal program formulas and got one for Ultra Fractal from Kerry Mitchell, but it's quite slow because of UF being a raster program. If anyone has one for vector fractal generators, please email me.

7/9: Lucian has done it again! This screensaver (1.2MB) does most of what I had in mind.

9/29: Added more historical info.

2003

6/4: I saw a The Learning Channel show about crop circles and briefly saw one that looked a lot like this image so I did some research and here's what I found: The International Crop Circle Database. Created on July 27, 2001 in Rybury/Tann Hill, England, the photographer, Lucy Pringle, comments "caption is now "A ring containing a number of ever descreasing sized circles arranged across its diameter." while crop circle researcher Freddy Silva comments: "poorly executed diagram of the principle of alternation" . :/ I hope he's referring to that particular crop circle and not the design itself...

9/21: Paul Bourke emailed me about a POV-Ray tutorial he created using this image.

11/23: Added some Javascript image changing.
11/30: Found a Yin & Yang image generator (at bottom) that can make images similar to circular infinity. Refined program request a bit to make it more clear.

12/31: Found (rediscovered, really) Steven Dutch's "Spherical Projections" (see "Stereographic Projection" ) which is an interesting variation on circular infinity.

2004

3/4: Added "regular network" image to Absolute Relativity. Created section titles.

4/18: Added "Cycles" application.

6/6: Roland van der Veen emailed me about a book called Indra's Pearls which supposedly contains circular infinity (but I didn't it see on the book's website, although is a close hyperbolic version). May 13, 2022 update: I have since found the book online as a PDF and on page 128 is circular infinity (referred to as "Circles and binary expansion").

7/2: Updated (and added a title for) news/updates with old news/updates (courtesy Internet Archive, which a link to was added at bottom).
7/4: Tweaked header.
7/8: Revised history. Tweaked news/updates (year headers, clumped site maintenance updates together). Fixed crop circle image link. Added Internet Archive link to Yin-Yang image generator.

2006

9/20: Replaced some origin section links with Wikipedia ones.

12/3: Fixed Rybury, UK 2001 crop circle link in 6/4/03 update. The crop circle database requires a login now. :/
12/6: Added "Zero/Infinity, Alpha/Omega, Beginning/End" to applications.
12/10: Applications: added "Games" and "Universe" and intro; reordered.
12/13: Added Wikipedia links for various things.
12/15: I've been working on recreating circular infinity in Mathematica and, with the help of some newsgroup users, have finally done it! The image under the blue Maya model (when clicked) is it; unfortunately, Mathematica can't anti-alias the lines so it doesn't look as good as the one I manually created in Photoshop. Mathematica is horrible to use (but better than MathCAD--which isn't saying much). I still want to create a vector version that can be infinitely zoomed in/out like a regular fractal...

2007

1/2: Michael Coleman has created an acrylic on wood panel version of circular infinity. Also on that page is a mention of a Mueller/Muller fractal and Global Scaling Theory (more specific info) and a diagram that looks a lot like circular infinity (only with 3 inner circles instead of 2) and is "The distribution of matter in the logarithmic space of scales in first approximation has the fractal dimension of Cantor dust, but is deformed hyperbolically in the proximity of a node point"--"logarithmic infinity" or "hyperbolical infinity".

Paul Bourke's (see 9/21/03 update) website is down. I emailed him recently about the POV-Ray version of circular infinity and he sent me all of the files to do it. Until his website is brought back up, the circular infinity page will be here. Also, this page is now located at http://tnlc.com/eep/circles/.

1/12: Updated absolute relativity.
1/30: Moved "History" section under "Origin" section and added Second Life info to "History", and vortex info to "Origin". Expanded Logic.

2/3: Fixed circles2 screensaver link in 7/9/02 news/update. On 1/22, inhahe, from IRC, emailed me programs that can supposedly "make circles roll around in the fractal circles image" but I have yet to install Python 2.4 and Pygame to try them out. Anyway, thanks inhahe!
2/5: I found this a few weeks or so ago, was going to add it to the page, but didn't for some reason. However, I refound this so I wanted to mention it here. San Graal Library - Alchemy, Magic, Incubation, Metatron, and Ancestors has an image on it called "nested binary spheres" which looks like a 4-level deep circular infinity but without the outer circle which supposedly represents the 3D world (reality). I particularly like the smaller "nested wave" image.

3/5: Added "complete graph" link to absolute relativity.
3/12: Replaced absolute relativity black-on-white-background image with white-on-black background and circular infinity GIFs with white-on-transparent background PNGs (the old one can be seen by clicking on the circular infinity image, which now defaults to horizontal vs. vertical orientation). Added crop circle thumbnail image to 6/4/03 news/update.
3/13: Replaced absolute relativity white-on-black background GIF with white-on-transparent background PNG. Forced page width to 1024 pixels. I saw a documentary, "Crop Circles - Quest for Truth" (2002), which featured an unnarrated segment on the circular infinity crop circle at 46:24-47:55 (watch it on YouTube).

3/18: I emailed Freddy Silva the other day, regarding the crop circle, and he replied: "The design is the classic image of the vibrating string which describes the frequencies of the music scale, by halving a vibrating string. I believe it comes from Robert Lawlor's book on sacred geometry. In any case, that crop circle was man-made (I actually was there to analyze it), but the connection is ironic since the real ones are made by sound (see my web site)." I couldn't find anything about a vibrating string that looked like circular infinity but I found a PDF of Sacred Geometry: Philosophy and practice (Robert Lawlor, 1982) and its section IV (p. 44) about "the principle of alternation" states (note π = pi, which looks like a narrower "n"):

The principle of alternation is expressed geometrically in the ancient Taoist symbol of yin and yang. The form of this symbol arises from two equal circles within the larger circle, the diameter of each smaller circle being exactly 1/2 that of the larger. The ratio of the diameter to the circumference of any circle is n; C/D = n.

At first glance the symbol suggests that the division of Unity (here taken as the large containing circle) is into two equal parts. Such a division results in a static equilibrium, without any possibility for growth. It is the asymmetrical division, as already demonstrated in the 1:J2 relationship, which creates proportion and thus the progression into form which we call growth. Later, in the Squaring of the Circle, we will discover the asymmetrical principle hidden within this symbol [in the assymetrical division]. But it is important to note in this context that the circumference of the smaller circles equals D/2 x π = πD/2. The sum of the circumferences of the two inner circles equals the circumference of the large circle (2 x πD/2 = πD). The figures show the continuation of this initial division carried through into 4 and then 8 divisions. This process of halving the circles can be carried on indefinitely; and at any point, when the sum of the circumferences of the smaller circles is totalled, it will still equal the original larger circle. This process can be taken to the point where the wavy line and the diameter become indistinguishable from one another, illustrating the paradox of the diameter becoming equal to the circumference of the same circle. Thus, like Theon's demonstration, this ancient diagram shows that at the origin and end all differentiation merges towards Unity.

3/19: Renamed Cycles to Cycles/Time and edited it.

• I've been researching more about vibrating strings and have found more images similar to circular infinity: Robert Fludd's mundane monochord (based on Pythagoras' philosophy of music, "the intervals and harmonies of the spheres", images of which can be seen on The Secret Teachings of All Ages (Manly Palmer Hall, 1928) and an interesting tri-sectioned circular infinity-cross-like image on Sacred Science's Seeker Sanctuary from Athanasius Kircher's book, Musurgia Universalis (which I found a PDF of that has the image on page 450 labelled as "Artis Magne Confoni & Diffoni" above it but that label is at the top of other pages, too, so that's probably not it; perhaps the next page's title: "Lib. X. Decachordon Nature.", which would be some kind of "nature 10-string harp-like musical instrument" or "10-part nature", roughly translated--but the image has more than 10 parts, that I can see; there are 9 circles on the 3rd level, plus the outer circle would equal 10...but I think that's pushing it; besides, that label is also on other pages, too (this book doesn't label its graphics very well and its index doesn't even HAVE the image listed!). Unfortunately, the PDF is entirely in Latin, which I don't know (and haven't had much luck translating it). If you know what this image's name means in English, please email me.
• Also, a new 2-hour Science Channel mini-series, Exploring Time, uses a variant of circular infinity to represent timescales. Its website has an interactive Flash animation that lets you zoom in/out on the circular infinity variant.

4/6: Added Internet Archive link for San Graal Library (which seems to be down) 2/3/7 news/update.
4/11: Wikipedia links for applications intro (and added "hierarchy").

6/4: I just found a reference to Jean-Pierre Garnier Malet's theorie du doublement ("theory of the unfolding" or "doubling theory") which, apparently, he claims to have come up with between 1996-2000 1988, interestingly... This theory is completely based around circular infinity.

6/15: Wikipedia consensorship deleted the "doubling theory" page I created but a backup is still no longer available. Agni: Way of Fire (Bruce Lyon, 2004, about Agni Yoga) has a "nine-fold manifestation of man" image, and various images regarding seven rays/planes, that are overlapping tri-sectioned (base-3 divisioned) circular infinities. Updated intro to reflect base divisions. I found Paul Bourke's new website (now here).

6/16: Stupid CSS tricks (overline, smallcaps, and image transparency). Found a much better chakra energy flow image ("chakra field diagram") that is pretty much exactly like circular infinity.

6/22: Sheesh...the chakra field diagram site fluctuates between being down and up (most likely an AOL server issue), but the Google cache still has it--and there's always Internet Archive.

7/23: Made a specific-resolution absolute relativity thumbnail so it doesn't get screwed up when scaled down; moved it to the right, too.

2008

5/3: On The History Channel's documentary television series The Universe, at the beginning of the "Cosmic Apocalypse" episode (season 2, episode 18) is a brief flash of an electromagnetic field (which circular infinity looks like, of course) right as the narrator says "bang" in "In the beginning, there was darkness...and then...bang...", implying this EM field represents the Big Bang. Changed electromagnetic radiation to electromagnetic field in intro.

11/7: Vincent Borja emailed me about Phun, a free 2D physics sandbox that can simulate circular infinity spinning. Vincent also created a circular infinity Phun file. Thanks Vincent!

2010

3/6: Fixed link to The Secret Teaches of All Ages in 3/19/7 update since About.com's version is down (and the Internet Archive versions don't have images).

6/6: Fixed links to crop circle database, comments, and image in 6/4/2003 update.

2012

9/23: Fixed nested wave image.
9/28: Searching Google for "circular infinity" yielded some "new" (to me) uses for this concept: something called the aether wave theory, renewable energy (scroll down the page, and other pages in the thread, for more examples), and rigid body motion rotation and translation.
9/29: Perusing the Google results, it seems that Walter Russell uses circular infinity in his 1959 "wave optics and space geometry series".

10/4: Updated absolute relativity with duality/monality and some minor rewording/expansion to cycles/magnitude/time. Added menu. Found Paul Bourke's new website yet again... Added some more links to history. Doing a Google search for "spherical infinity" I came across:

• Edward Ward's laser-machined engraving of circular infinity (not spherical infinity as he calls it), 12 July 2011
• Peter Hanley (and others)'s Processing sketches, 5 July 2010

10/5: C. Om has a good expansion on absolute relativity. Bulleted absolute relativity visual representations and added a new one: all fractals. Added symbols to, and minor rewording of, Zero/Infinity, Alpha/Omega, Beginning/End. Added intersecting circular infinity to top image (click to see). Reginald Brooks divides up the "geometry of music and color" monochord using circular infinity (most likely based off Robert Fludd's mundane monochord work).
10/20: Did a Google search for "torroidal universe" (after watching Thrive) and found circular/spherical infinity used on some Mormon Transhumanist website about "consciousness is a fundamental quality of absolute space" (whatever that is) and draws connections to zero-point energy, the chakras, tai chi, reincarnation, etc.

11/25: Did a Google image search for "fibonacci" and found some Fibonacci spiral art relating to circular infinity at Kean Walmsley's Creating Fibonacci spirals in AutoCAD using F# and Sidart (site can take a while to load).
11/26: Renamed "Logic" to "Logic/Math", fixed misspelling, added some Wikipedia links. Circular infinity shows up as Jason Davies' prime number periodic curves (article about it) and as a Poincaré disc/disk (hyperbolic circular infinity).

12/1: Fixed Agni: Way of Fire links.
12/11: Moved circular infinity image out of text in order to facilitate better stereogram effects. To see it, cross your eyes and stare in the center; by crossing your eyes more and more, you can see more levels.

2016

3/11: Holy crap, An update! I was perusing YouTube and came across a video called "The Sacred Geometry of The Universe" which, at 1:13:55, shows a sine wave version of circular infinity ("sinular infinity"? sounds...sinful--blasphemy!) The video is a compilation of Spirit Science videos (Spirit Science 22 (part 2) ~ Cycles N' Sine Waves) which explains in more detail a lot of what I mention in the Applications part of this page; good stuff (even if the video does get a little cheesy at the end). In the same YouTube video, at 1:24:26, circular infinity is seen at the top of a spiral painting by Adrian Kenyon, explained as "The last part of the spiral is the 7th dimension and the realm of God. An infinite number of multiverses dividing like cells in an embryo. An ultimate omnipresence as everything that is contained in all of these multiverses is a part of God's consciousness." In the YouTube video of the artist explaining an unfinished version of the painting, at around 8:03 he says "these forms are slices through 4 dimensional spheres and represent an 'infinity of infinities'".

3/12: A Google search for "hypersphere" led me to an image (near right) (on Annandi Astrology Blog - May New Moon: The Path to Becoming) which is close to the triple intersection of circular infinity you see if you click the main image at the top of this page (and far right).

3/13: Googling "circle into wave", I rediscovered the aether wave theory mentioned in a September 28, 2012 update and found, in the comments (posted in November 2015 by Zephir), an HTML5 fractal animation of circular infinity forming out of waves (slower, 1000-pixel, 10-level version). Zephir also found a link to a 2002 webpage, Research on Networks and Random Walks (Stefan Boettcher with Bruno Gonçalves), which introduces network classes that resemble circular infinity.

6/15: A "squaring the circle" Google image search yielded the image at right found on "CodeX4 and the Reconciliation of Science and Religion" by Raphael (Kachina2012) (under "CIRCLE and the SQUARE"--no direct link) where a variation of the yin-yang symbol (and approaching a 2-axis-intersected circular infinity) can make a lauburu, or swastika-like symbol.

2017/9/18: I was looking up how to show 2+2=5 to someone on Twitter when I came across gray code which, in its "pretty pictures" section (sorry, no direct link but it's about half-way down the page) has a picture of what looks like half of circular infinity:

2018

7/20: A Google search (for "geometry code") led me to Clusters: Vortex Field Guide which has images of circular infinity by Barbara Hero and Bruce Rawles from his book Sacred Geometry Design Sourcebook (c. 1997). I can't believe I haven't come across this book until now but I think I will try to contact Rawles and ask him when he discovered his version of circular infinity. As for Hero's image, I found variations of it in The Sub-Conscience Speaks; with Lambdomas (PDF, 1990, pages 2,19-22,28), a book she illustrated dated June, 1988. Also in Hero's book, Lambdoma Unveiled (The Theory of Relationships) (PDF, 1992, pages 91,102,103).

7/21: Fixed crop circle image/link yet again.
7/25: Added authors/dates to some March, 2007 updates. I created a circular infinity Pinterest board to make it easy to collect images.
7/26: Fixed links in 2002 and 2003.
7/27: A Google search for "css fractal" led me to a CodePen page, where I started trying to create circular infinity out of CSS and then figured I'd do a search for "circle fractal" and, lo and behold, I found circular infinity in alternating filled form (animated too) by Sufien Tout. I forked it and commented out the animation. A Google search for "circle fractal" led me to Remember Java - Cantor circles which has some Java code to make circular infinity and a cool animation.
7/30: A Pinterest ("more like this") search led me to "Yin and Yang Fractal" by Simon C. Page (which isn't actually a yin-yang--no dots--but a variation of circular infinity) and, from Google, other yin-yang-like versions: "Fractal Yin-Yang" by Nathaniel Hellerstein and "Binary Ratio Diagram - YinYang variation" by Bartosz Majczak (aka Bartomon); more variations.
7/31: Fixed doubling theory link and added earlier date.

9/16: Moved 2D/3D circular/spherical infinity names to beginning of intro paragraph for context. Tweaked image animation to use more CSS and less javascript.

2021/5/4: Fixed Athanasius Kircher - Decachordon Nature image link.

2022

3/31: Tweaking mobile ("responsive web") design of the page; it now fits on smaller screen widths.
4/1: Wrapped text circularly around top circular/spherical infinity images.
4/2: Condensed single-entry news/updates dates to format year/month/day. Reduced margins on horizontal rules and headers. List markers (such as the bullets/discs in Applications: Absolute relativity) don't wrap correctly around CSS shape-outside:circle() styles, for some reason. Unfortuately, this happens in both mobile and desktop versions of Google Chrome.
5/12: Browsing Instagram, I found an actual physical application of circular infinity as a vise.
5/13: Updated 6/6/2994 update after noticing a dead link (one of many on this page, I'm afraid).
5/16: Fixed first circular infinity image so it cycles through when clicked for mobile users.
5/28: Added horizontal+vertical intersected circular infinity to clickable first/main image sequence.