TOPOLOGICAL FIGURES, COOK-MÖBIUS CROCHET AND NOBLE SHAPES
YOU HAVE MADE THE BODY OF WORK IN MY PREVIOUS POST, LEARN ABOUT QUANTUM PHYSICS THROUGH CROCHET, it is a paper Möbius strip grown on to row 1, complete. The paper strip is equivalent to the crochet Starting Chain, the first layer of safety pins is equivalent to Row 0 of Stitch, and the second layer of pins is equivalent to Row 1 of Stitch.
TOPOLOGICAL FIGURES, COOK-MÖBIUS CROCHET AND NOBLE SHAPES
After the beginners' crochet chapters, the bodies-of-work we make will be crocheted Möbius twists/Möbius Rings, and later on Cook Twists/Cook Rings. They will be actual manipulable topological symmetrical figures. You can stretch them and pull them about, but they will remain essentially the same.*** As you grow them on Row by Row, they will get bigger and fuller, as well as smaller, because tighter. As with your paper strip grown-on you will need to pull the working edge up and out of the core gap (the twist) in order to stitch the next stitch, and eventually there will be so much growth that the core gap gets full, and it is harder to pull the working edge out to add another stitch. Before that event, your Body-of-Work will be arranging and folding itself into beautiful and mysterious shapes, which I call Noble Shapes. I predict you will play with them as they get bigger, and here might be useful a reminder of the first rule in crochet, which is to Always Keep The String Behind The Work!
***However, if you use an even number of (half) twists 4 etc, and then grow the "mobius ring" on, eventually the twist and the core gap will disappear! It will look like a simple cylinder or a bowl with a complicated bottom. Its form will change. Another confounding fact: Things Disappear. Whether this means such an object is not topological, even though it seemed topological before you made it bigger, I cannot say.
WHAT IS TOPOLOGY?
According to the OED, "Topology is the branch of mathematics concerned with those properties and surfaces which, unlike shape and size, are unchanged by any continuous deformation (i.e.excluding breaking and tearing) and hence with those abstract 'spaces' (sets composed of points and subsets of points) which are similarly unchanged by certain kinds of transformation". Topology is a new branch of mathematics, created for Mobius Twists and other similarly one-sided and one-edged figures.
NOBLE SHAPES
These days, complicated topological figures are still mostly theoretical and mainly exist on a computer screen, with the exception of a Möbius Twist, which is an actual (i.e. not just theoretical) topological figure much used and discussed by children, as well as quantum mathematicians, engineers and theorists. As far as I know, however, I am one of very few with an interest in quantum theory who 'grows' Möbius strips 'on' and has developed a studied method for creating different kinds of topological Objects / Twists from a crochet chain by studying the gaps, rather than the stuff. I call this growing-on of Möbius twists Cook-Möbius Crochet, and the resulting topological bodies-of-work I call Noble Shapes. I have names for some Noble Shapes that appear at subsequent stages of growth, e.g. a Cornucopea, an Atom, a Shell, a Tipi cover, a Slipper, a Boat, a Spaceship, a Recycler, a 4-way Pole Connector... There are different variations of Möbius ( and Cook) Twists, depending on what you 'program' into the chain at Row Zero of Stitch, and on whether you use one, two or three 'sides' of the Starting Chain, and how many and what kind of twists you add. The resulting bodies of work from these variations automatically form different Noble Shapes. You can also be creative by applying a continuous formula after Row Zero is complete, i.e. by consistently adding x number of stitches in every y type gap of the working edge or by similarly skipping stitches, or by using a combination of the two. Another topological figure is the double helix, which represents the strands of Earth's Life's DNA. The Double Helix and The Atom are based on Möbius Twists.
TOPOLOGICAL FIGURES ARE VERY USEFUL
Quantum scientists use topological figures because they help explain the quantum realm of further dimensions. We conventionally perceive three dimensions plus time-- (our dimensions are length, width and height --think of how you measure the dimensions of a piece of furniture you are hoping to buy and fit into your home, or think of it as giving the coordinates to meet up with someone, you need to describe the length ,or longitude on a map, also the width, or latitude on a map, plus the height-- 57cm? the basement? the International space station?, and also, but only if you are giving coordinates for a meeting, you need to describe when in time you will meet. 9.a.m? Next week? In 10 years?Thus, Time is considered a dimension, though of a different kind,). Quantum physics and mechanics use the fact that there are more than 3 spatial dimensions (maybe 10) to make fabulous and powerful tools eg. quantum computers and nano-technology, but these other dimensions are still a mystery and they are unsure how many more of them may exist, nor where they may exist, ha ha, nor whether there are dimensions where time does not exist, or space does not exist. Their best guess is that they are folded into each other. A strange thing about quantum physics is that no one really understands it, yet we can make calculations and machines based on and using it.
YOU DO NOT HAVE TO UNDERSTAND IT IN ORDER TO DO IT!
As with quantum physics itself, you do not have to understand Cook-Möbius Crochet in order to do it. Rather, because you are doing it, you gain an understanding. You have started from scratch and grown your topological body-of-work, stitch by stitch and that is what I mean when I say you will gain an innate sense of the quantum realms through Doing Cook-Möbius Crochet. You know that body-of-work intimately, every single step and stitch of it, because you made it with your own two hands. It will fold itself.
ANCIENT KNOWLEDGE
Humans have been making and using Mobius twists ever since we started using string to craft things, which I reckon is as long as there have been humans. Twists happen easily and not always intentionally, as anyone who set out to crochet a simple rectangular scarf, only to find that it somehow got itself into a permanent twist can testify to. Or, indeed, anyone sewing or knitting who got their string into a ruinous knot. Knots are Möbius twists, and Nautical Knots are an art and a science of their own. Drawing Knots is also an art and science of its own, kept during the dark ages in Ireland. Alice in Wonderland is famously a person who fell asleep while knitting and went down the rabbit hole where reality and dimensions are changeable and very very strange. People forget that Alice's Adventures all started with knitting. Perceiving quantum realms of further dimensions is already well-entrenched in the human consciousness. I believe consciousness itself is a dimension, which is proved by the famous Schrodinger's Cat experiment, and the acceptance by physicists that the very act of observing something changes it, and can influence outcomes. When we dream, we are in a dimension where time and space and causality do not exist, or at least not like they do when we are awake. We spend a third of our lives sleeping, so we are all already intimately acquainted with the quantum world without even trying. Anyone of my generation (1960's) who watched the original Star Trek television series has witnessed that the fabulous machines and capabilities that were pure imagination in Star Trek have mostly come into reality. We still have yet to actualise a powerful Drive for a spaceship, and we are yet to discover how to manipulate Gravity -- Qantum Theory, Mechanics, Mathematics and Chemistry has not advanced enough for that, but it will. In the meantime, we might not even need those innovations: meditation practitioners claim they can travel and communicate without even moving their body. Meditating opens the gate in your brain between your subconscious and your conscious mind. It connects your everyday awareness directly to your quantum awareness. Why do you think Buddhas are smiling? It is all quite hilarious. Meditation also heals the body faster than sleep. We have all come across the phrase "the power of the mind", well the mind might be the most powerful thing of all.
Coming Soon! My next posts of The Möbicorikon: The Crochet Starting Chain And Gap Maths; and: Begin Crochet
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