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| <!DOCTYPE html> | ||
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| <meta name="viewport" content="width=device-width, height=device-height, initial-scale=1"> | ||
| <title>Fractal mazes</title> | ||
| <link rel="stylesheet" href="../style2020.css?v=1.3"> | ||
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| <div class="sidenav"> | ||
| <img src="../_site/site_cornersmall.gif"> | ||
| <p> | ||
| <img src="../_site/logo_white.png" height="250px"> | ||
| <a href="../index.htm">home</a> | ||
| <a href="../new-faq.htm">help</a> | ||
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| <div class="mainbody"> | ||
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| <div class="page-intro"> | ||
| <h2>Microchip fractal mazes (2024 collection)</h2> | ||
| <p> | ||
| This page offers a selection of fractal mazes inspired by the works of Mark J. P. Wolf (2003) | ||
| and Ed Pegg Jr (2011). The collection starts with four hand-crafted mazes first published in | ||
| early 2021, now ported to PS+ and updated for faster game play. | ||
| The new set of designs for 2024 have been derived from computer-generated data provided by | ||
| Erich Friedman in late 2021, adapted to demonstrate a variety of new fractal maze-forms. | ||
| </p> | ||
| </div> | ||
| <div> | ||
| <p>Use cursor-keys to move the spark along the wires, the spark will keep moving until it | ||
| reaches an unvisited goal or a decision point. Goals are indicated by orange squares and | ||
| can only be harvested on the home-level of the maze. Use ESC for level select.</p> | ||
| <p> | ||
| Mazes 1-4: [2021] Bi-directional, single-goal, single-chip (the 2021 collection).<br> | ||
| Mazes 2-?: [2024] Bi-directional, multi-goal, single-chip.<br> | ||
| Mazes ?-?: [2024] Bi-directional, multi-goal, dual-chip.<br> | ||
| Mazes ?-?: [2024] Bi-directional, multi-goal, split-chip.<br> | ||
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| Here is an image of... | ||
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| <div class="page-note"> | ||
| <h4>Development notes - History</h4> | ||
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| <p>The Wolfram maze (2011) is an 8-pin maze, in a 22222/33 configuration. The 8 | ||
| edge-connectors (pins) offer 8 inner (downward) and 8 outer (upward) termini. | ||
| Of the 16 termini, 2 sets of 3 are connected to form the two decision points. | ||
| The other 10 termini are plumbed in 5 pairs with no decision points. | ||
| In 2011 I was intrigued by the depth of the recursion required to solve | ||
| such a minimal fractal design, but I also noted that if I could only | ||
| recurse outward from the start point I could reach the goal in a single move. | ||
| </p> | ||
| <p> | ||
| In 2021 I eventually started exploring some bi-directional Wolfram-style mazes | ||
| of my own. I implemented them in rudimentary puzzlescript, and shared them at | ||
| itch.io. Later the same year Erich Friedman kindly started exploring other | ||
| configurations of 8 and 9 pin mazes, specifically configurations 22222/33 (8-pin) | ||
| through to 333333 (9-pin) with most effort spent on 22/3333 (8-pin). | ||
| The data taught me a lot about what was and wasn't possible with single-chip | ||
| fractal mazes. Ultimately provoking me to comment (in an email in | ||
| Jan 2022) that fractal mazes are <q>endlessly fascinating in their ultimate | ||
| dullness!</q>. I needed to take a breather. | ||
| In Jan 2024 the dullness finally dissipated and I was tempted to take | ||
| another look at Erich's data, resulting in this, the 2024 collection. | ||
| </p> | ||
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| <h4>Development notes - Implementation</h4> | ||
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| <p>Puzzlescript is an ideal engine for implementing many grid-based logic-mazes, | ||
| however for fractal mazes it poses a few challenges. With no stack, and only | ||
| primitive mechanisms for counting how do you implement a potentially infinite | ||
| recursive maze? The short answer is... you don't. | ||
| Setting a generous (but finite) recursion limit avoids one set of problems, | ||
| but if a fractal shape can contain multiple copies of itself, then it is very | ||
| difficult to track current state without a stack. So to keep things simple | ||
| I decided to side-step the latter problem completely. Each | ||
| fractal chip is unique and contains a maximum of one copy of itself (or | ||
| another chip) following closely the simple model of the Wolfram maze.</p> | ||
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| <h4>Development notes - clickmazes variations</h4> | ||
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| <p>One key benefit of non-repetition of the fractal shape is there is no | ||
| ambiguity with recursing "upwards" from the start level, thus all mazes | ||
| featured here allow you to explore in both directions. But given this | ||
| restriction what other variations on the fractal maze are possible? Most of | ||
| the more complex fractal mazes you find online rely on repetition of | ||
| the same fractal shape within itself. | ||
| My first idea was to make the mazes multi-goal (by placing a separate | ||
| target on every home-level path). Then to start redistributing the | ||
| available microchip "pins" to form dual-chip and split-chip mazes. | ||
| Dual-chip mazes consists of two non-identical chips each containing a | ||
| copy of the other (to break parity direct connections between the dual-chips | ||
| are also permitted). | ||
| Split-chip is a further variation where the outer edge of the fractal | ||
| chip is an amalgamation of several non-identical inner chips. So 4 inner | ||
| chips, each with 3 outbound termini, yields an outer chip with 12 inbound | ||
| termini. These fractal mazes start to feel a little "cursed" but are they | ||
| fundamentally harder? Let me know if you decide. | ||
| </p> | ||
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| <div class="page-note"> | ||
| <h4>Links and related pages</h4><p> | ||
| <a href="https://demonstrations.wolfram.com/FractalMaze/">The Wolfram maze </a>(Ed Pegg - 2011)<br> | ||
| <a href="https://www.mathpuzzle.com/18Nov2003.html">Mathpuzzle.com archive of the earliest fractal mazes</a> (Mark J. P. Wolf - 2003)<br> | ||
| <a href="https://www.puzzlescript.net/play.html?p=7c6fe7b9f6397c0006da099d7f015cea"> Clickmazes Microchip fractal mazes V1 in PS</a> (2021)<br> | ||
| <a href="https://freethoughtblogs.com/atrivialknot/2023/10/18/solving-fractal-mazes/">Solving Fractal mazes</a> (freethoughtsblog)<br> | ||
| </p> | ||
| </div> | ||
| <hr> | ||
| <div class="page-credits"><p> | ||
| <b>concept </b> - Mark J. P. Wolf - 2003<br> | ||
| <b>maze designs</b> - © Andrea Gilbert and Erich Friedman 2021-24<br> | ||
| <b>puzzlescript implementation</b> - © Andrea Gilbert - 2021-24<br> | ||
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| </p> | ||
| </div> | ||
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