Saturday, April 4, 2015

Clifford Torus Turns Inside Out

Today I ran across this lovely gif of a "Clifford Torus" on wikipedia, created and posted for use by Jason Hise, using Maya and Macromedia Fireworks.

Meditate on this image for a while and see if you can track a patch of surface through all its permutations.


  1. Hi Cyd, Bonjour ...

    My main thoughts this week ... The first one : We don't know how the sphererical shapes of planets and stars are filled. Looking at it on the surface, they may look spherical :
    But, how this sphere is filled ? Is it homogeneously filled or is it filled like a torus ? Nobody knows.

    The first thing is : nothing is perfect. Something which looks spherical on the outside may not be an idealized, homogeneously filled sphere. For example, the perfect fluid ( ) is used for theory, but never corresponds to reality ...

    So, how is our earth filled ? There are some videos on youtube which talk about "hollow earth hypothesis". If there is really a hole in the earth, then it may fit the toroid-volume-filled view which explains the shape of pumpkins and the interior of oranges.

    This hollow earth hypothesis : have you heard about it before ?

    meditating on the interior volumes,

    1. Hi Tony. Take a look at this blog article on the shapes of protons from 2011: . Since the large and the small are both scaling phenomena of self-similar fractals, it is reasonable to wonder if whatever goes for a proton, could go for the earth, as well.

      The hollow earth hypothesis is old. Read about it when I was a kid; I'll revisit the subject.

      I've been reading on wikipedia all of the articles regarding fractals and fractal geometry. Your notion that trees and spheres are material expressions of geometric ideals is a well trod subject in fractal articles. I'll try and get an article about that up soon...

  2. Meanwhile, have you tweeted your friends about the blog and about my book? Trying to muster up a word of mouth campaign for the new $10 edition. :-)

  3. Hi Cyd,

    Seems you already knew about all this - hollow earth hypothesis - I've found this only recently ...

    Quote from your article :
    "But then I asked him if the spherical protons might not look spherical on the outside, but possess toroidal flow energies within their borders. He said, "yes, that could be the case."

    => here too, I see that you have already been through this "inner-outer-problem" 4 years ago ! I thought I was the only guy on the planet questioning about how the interior of earth is ... And if I can relate it to some well-known geometric pattern such as concentric hollow spheres, and from there come to your views by deduction.

    Otherwise, I'll speak about your book to my friends. All of them are very busy with jobs and children ... For someone to ponder on the subject, there must be a mind detached from personal worries. How can a person concentrate on a subject like this if he/she has no time to think ?

    today, I'm dreaming about

    As you can see, I'm a big dreamer, going back & forth betweeen colorful imagination & black-and-white, binary rationality ;-)


  4. We used to live in Mt. Shasta City, California. There are people who live in the Mt. Shasta area who believe the volcanic funnel of the mountain is a tunnel to the hollow earth inside. The WWI flying ace, Rickenbacher I think was his name, flew over the funnel and saw the sun inside and claimed to have no doubt about the hollow earth theory.

    Other people in the area believe that a race of people they call "Lemurians" lives deep inside Mt. Shasta, having retreated there a long time past. They think the Lemurians are related to the lost people of Atlantis.

    There are also people who believe that the undead yogi Babajii lives in the mountain and comes out once a year to bestow blessings.

    Jules Verne wrote a classic sf novel about all of this called Journey to the Center of the Earth.

    I do know that the solid earth is only an hypothesis because core samples can only go so far. There was recently a science article about some major change in the solid earth hypothesis modifying the nature of the center of the earth.

  5. Hi Cyd,

    Very interesting to know about all this about HEH (Hollow Earth Hypothesis) ... I talked to one of my friends about toroids, spheres and your blog. He talked me about .. That's what he knows about toroids.

    Since a moment, I got more and more into "tree observation" and what I think is that there is a lot of variety in trees. Due to spring time here in France, I also got interested in flowers : their colors especially. As Alan watts' said :
    “The wild geese do not intend to cast their reflection. The water has no mind to receive their image.”

    I think flowers too do not intend to be in such a geometrical shape or even planned to have such gradients of colors. They don't know about themselves. Same thing for Chameleons : they don't change their color on purpose according to some article.

    If I go deeper with this reasoning ... Humanity wasn't supposed to exist and scientific laws are not intended to be discovered. There is a universe out there and our understanding of it changes all the time.
    May be, we humans would better pray God ;-)

  6. I just finished reading a book I should have read years ago. I have to recommend it to you. It's called "Chaos" by James Gleick. (I'll write a book review next time.) The author really makes sense of chaos theory and fractals and exactly, MATHEMATICALLY, very, very few formulas result in all of the variety we see around is. So, it's not even so radical any more to say that a very simple, fractal formula is responsible for a variety of manifestations, from stock futures to cell division to weather. Turns out the same exact formulas describe tons of things that aren't even related to one another in any way other than a particular pattern of replication, division, and arrangement.

    As far as the Simple Explanation goes, I'd say the chameleon may not consciously choose to change color, but the chameleon's skin units of consciousness do. In my Simple Cosmology, there is a unit of consciousness attached to every decision-making entity, and since everything in this universe is tasked to decide something, even if it's just to go left of right or spin up or down or maybe it's to work a job in Paris and live outside the city, no matter what the level of decision-making, a unit of consciousness literally makes the decision. Have you read the blog article called A Simple Fractal Model of the Conscious Universe?

    Enjoying talking with you Tony. Thanks for your interest and your forwarded links..

  7. Hi Cyd,

    Nice answer. I was going to stop all thinking and pray God ;-)

    Thanks to you, I now understand that simple equations of Chaos theory may lead to all this variety. As I said, I felt overwhelmed by colors in flowers : it was way too much to understand what was their origin. And then searching on internet wether there is some sort of Alien intervention, I came across some scientific articles stating that our DNA is not explainable through simple evolutionaty process but there is strange arithmetic underneath well hidden in the DNA which uses zero, fractions ..

    Ok, I won't stop my enquiry on nature. I'll read "Chaos" by James Gleick. Prayer is good though.


  8. Hi Tony. Yes, prayer is good. Meditation is good. Glad you know this. Exciting stuff.