Circle Puzzler's Manual Part 2
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1 Circle Puzzler’s Manual Part 2 By Doug Engel Front page 1 2 Circle Puzzler’s Manual Part 2 By Doug Engel 2019 Copyright © 2019, Doug Engel, Littleton, CO, USA Being an update to Circle Puzzler’s Manual, ©1986 https://www.jaapsch.net/puzzles/circleman.htm https://www.puzzleatomic.com/circleman.htm Printed and online in United States of America https://www.puzzleatomic.com/circlemanpart2.pdf Available free online but hard copies must be purchased. Email [email protected] for hard copy availability First printing rev. zero Dec. 2019 Quotes and graphics by other authors included in this work are copyrights owned by those authors or joint authors. Links are not guaranteed to remain active. We will attempt to note or remove inactive links in updated revisions or releases. 2 3 Dedicated to my wife Neva and all the puzzle designers, solvers and collectors. Special thanks to Jaap Scherphuis who originally put CPM online making it widely available. Thanks to Bob Hearn for his super code, producing the first circle puzzle fractals and for his numerous talks clarifying Rc (critical radius) and how these fractals are produced. Thanks to Jason Smith for his Gizmo Gears Deluxe Render showing fractal movies and his fractal research. Thanks to Brandon Enright for his fractal code and his extensive mathematical analysis of Rc. Thanks to Andreas Nortmann for putting the Gizmo Gears jumbling question on the forum after Battle Gears failed and I suggested Gizmo Gears might be worth a try. Thanks to Oskar van Deventer’s energetic input and puzzle innovations and alerting me. Thanks to Bram Cohen for first suggesting the investigations into jumbling puzzles. Thanks to Carl Hoff for many enlightening comments. Thanks to Mathologer, Burkard Polster for his you tube’s and to Roice Nelson for his Magictile. Thanks to many others on the Twisty Puzzle Forum for the lively investigation, over 450 posts, that produced many insights into these fractals. Including: David Litwin, Guilty Bystander(Landon Kryger), Jared, Kattenvriendin, Rubikcollector123, Will_57, VeryWetPaint, Eric, KelvinS … Thanks to Martin Gardner (Hexaflexagons), Tom Rodgers (G4G), Jerry Slocum (IPP), Tom Cutrofello, Nick Baxter, Scott Morris (Omni), Erno Rubik, Doug Hofstadter, … Acknowledgements Twisty Puzzle Forum …, Wikipedia, … 3 4 Circle Puzzler’s Manual Part 2 (A tale of circle puzzle jumbling) Definition of some of the words used with circle puzzle fractals (read only if you are unfamiliar with any of these terms). You can return here as needed. Fractal: A self-similar figure with fractional dimension. For instance, the world famous Mandelbrot set is a fractal. You can zoom in on various parts of it and each zoom looks fairly similar to the previous one. Doctrinaire: A twisty 3D Rubik type puzzle or flat circle type puzzle where each move of one group of pieces allows you to the chose the next possible move of a group of pieces without restriction. Bandaged: A twisty puzzle where at least one pair of adjacent pieces are attached or glued together. In order to move when the glue would prevent a move you want to make you must cut the two pieces apart, then make your move. Jumble or jumbling: This is the key concept that leads to the Gizmo/Hearn/circle puzzle fractals. It happens when you find a puzzle where every time you cut through a bandaged area you end up cutting as piece of it again until an infinity of pieces are the result. This in turn can produce amazing and beautiful fractal graphics and crystallographic effects, depending on the rotation generators used by the computer program (code) and various other parameters used to generate the images such as how points are colored and so forth. Critical Radius R, Rc, Rcx: The critical radius happens when the radius of the two circles becomes large enough to cause jumbling. It has been proven in the forum, to stay critical from then on as the radius grows larger all the way to infinity. I have added Rc as a parameter for the critical radius and Rcx >= Rc, where Rcx can be Rc or any radius larger than Rc. Surd: Used in this manual to mean irrational. N: N is the smallest possible rotation you can make for a group of pieces. For instance if N is 4 then the smallest rotation is 360/4 = 90 degrees. Gizmo Gears: The two circle puzzle showing both 3 way and 4 way symmetry. The question was posed “Are the Gizmo Gears Jumbling” on the forum: http://twistypuzzles.com/forum/viewtopic.php?f=1&t=25752&sid=94f858d2f15227ed1a3b97 e453f75f20 4 5 Table of Contents Motivation(this page) Introduction to some Gizmo Fractals Bob Hearn’s code About the Fractals A formula for the number of pieces in a single period binary circle puzzle A circle puzzle derived both from a surd and a Gizmo render. A surd 2 infinity clock Polygonal puzzles and tessellations derived from circle puzzle pieces Mono-polygon puzzles Review of a selection of the latest mostly planar twisty puzzles Slide Rules Deep Cut Slide Circle puzzles Palette Puzzles Flatland A property of flatland puzzles Set theory of circle puzzles What is the meaning of all this twisty stuff? Twisty puzzles as gears Fractal tiles for smaller N High order code puzzles Mathologer demo of magnetic cubes Some quotes taken out of the fractal forum Page 1 Page 2 Page 3 Page 4 Page 5 Page 6 Page 7 Page 8 Page 9 Page 10 Page 11 5 6 Some of the Motivation for this Update CPM classifies various kinds of circle puzzles. It did not classify all possible types that can be designed (unlimited!) but did suggest various sets that some elementary designs are elements of. CPM is a way by which one can specify the major geometry aspects of some given circle puzzle. There are some examples of this in CPM. One feature of the manual was the idea that symmetrical deep cut (two or more pieces in the circle intersections) circle puzzles can generate an infinite number of pieces if the intersecting circle radii are increased without limit (increased to infinity). For instance, an illustration of hexagonally symmetrical circle puzzles, in CPM, with radii increase, show rapid increase in the number and shapes of the pieces as the two circles approach each other (R=radius of the two circles increases). The surprising thing is that the number of different congruent pieces increases very slowly. At infinite radii it was called an infinite or perfect black circle in the CPM: https://www.puzzleatomic.com/images/circleman/p35.gif. Since then at least one designer, Oskar van Deventer has produced some designs not classified in CPM. Any errors, technically, editorially, graphically, network links, and/or in mentioning other’s efforts and opinions, the fault is mine. I will try to make corrections in a revised edition. For instance, my calling N=infinity might be objected to. The Gizmo gears is designed to have two N’s, N=3, and N=4 or N=(3,4), (can Gizmo Gears (3,4) restriction be turned into a fractal?). Gizmo has both triangular and square symmetry in a bandaged mode. Gizmo with unbandaging attempted is 3x4 or N=12. Thus, saying N=infinity if theta (here theta=minimum turn angle) is irrational seems correct. It drives home the point that if N never closes on itself it must become very small. Theta irrational makes it possible to have single circle fractals, or (pseudo fractals?) in an infinity of different ways which is explored below. When an infinity of cuts has been made based on irrational theta, then N can be turned an infinitesimal amount before the next possible turn. Bob Hearn mentions N being irrational on the Gizmo forum “… of course we'll always get jumbling when N is irrational.” In this case he means the same thing, that the rotation angle, is based on an irrational number. 6 7 This brings up another controversy that was on the forum, “can any of these really be considered fractals?”. I believe they are all fractal-like, even the N=2,3,4,6 ones with no critical radius, except infinity, because at a large radius the pieces become increasingly numerous and complex and follow repeating patterns, in a time based or progression based manner. To see the pieces, you would need to use magnification of an infinitesimal but finite(!) area, for the N=2,3,4,6 ones when R ~ infinity. Infinity is a paradoxical concept when you try to tame it. If you are not familiar with N, R, jumble, etc. read on or refer to the word definitions area. Jumble means a puzzle that cannot be completely ‘unbandaged’ by cutting thru the ‘bandages’. Bandage means two pieces ‘glued’ together so you are prevented from making a move until you cut thru the piece, creating two pieces. You cut thru ‘bandaged’ pieces to try to make it so that a move is always possible and therefore unbandage it. If, after continuing to make cuts, it is not unbandaged then it continues to forever be cut into more pieces, and it is said to jumble. The word jumble and jumbling was first introduced by Adam Cowan who designed the Bevel cube, here is a link by Bram and Oskar: http://twistypuzzles.com/articles/other-twistypuzzlesthatjumble/ Bram probably introduced ‘bandaged’ and ‘bandaging’ and perhaps ‘doctrinaire’ for non- bandaged puzzles. I may sometimes use the word tame for doctrinaire. Bram, after many discussions with twisty puzzlers provided a mathematical definition of jumbling, when the pieces turn to Bram’s dust. Here is what David Littwin has to say “…As for history, the feature was first described by Okamoto when noting the motion of the helicopter cube: https://www.twistypuzzles.com/forum/viewtopic.php?p=48762#p48762 He and Adam Cowan both built this puzzle independently about the same time and Okamoto wasn’t sure if Adam’s jumbled.