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I want help reading a scientifficc paperOf course! Please provide the URL link to the scientific paper you'd like assistance with, and I'll help you extract and understand the information from it. If you don't have a URL but have the paper in PDF format, you can upload it directly, and I'll assist you further.How do I upload itTo upload the scientific paper in PDF format, follow these steps:

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Once you provide the doc_id, I'll be able to assist you further with the content of the paper.Can you explain how to draw tiles and metatiles like I am a 5 year old. This is the scientrific paper a8d0a02f-8b9a-4002-ac37-c2f8f0ea9126{ "doc_id": "a8d0a02f-8b9a-4002-ac37-c2f8f0ea9126", "query": "tiles and metatiles explanation" }{"results":[{"doc_id":"a8d0a02f-8b9a-4002-ac37-c2f8f0ea9126","text":"preprint\n33\nTheorem 5.1. In any tiling by the four metatiles, after such bisection of P and F metatiles,\nthe metatiles fit together to form larger, combinatorially equivalent supertiles, thereby forming\na substitution system. The tiling by the supertiles has the same symmetries as the tiling by the\nmetatiles.\nFigure 5.2: Bisection of tiles P and F\nThe bisection of tiles is not strictly necessary, in that the bisecting lines can be arbitrary\ncurves—and, in particular, can go entirely along one side or other of the F or P tiles (keeping\nthe same end points), effectively allocating an entire tile to one of two neighbouring supertiles.\nHowever, the bisected tiles are convenient for proving that the supertiles obey matching rules\nequivalent to those of the original tiles. In particular, bisection causes adjacencies between\nsupertiles to be more clearly encoded in the boundaries of the supertiles themselves, without also\nrelying on information about forced tiles that are not part of the supertiles. In some situations\nit may be more useful to assign whole tiles to supertiles at every level of substitution, with\nno bisection. For example, these whole tiles may be more convenient for analyzing sizes or\ngrowth rates of patches in the inflation process. If needed, we can define a symmetry-preserving\nbijection between the supertiles shown here and any alternative choice of supertiles that avoids\nbisection.\nDiagrams for analysis of cases should be interpreted as follows. There are some unnumbered\ntiles that define the case being considered, then some numbered tiles that are forced in the se-\nquence given by their numbers. If it is then necessary to split into multiple cases, the position at\nwhich multiple choices of tile must be considered is marked on the diagram with a filled circle,\nand there are then separate diagrams for each choice (on which the previous forced tiles are now\nunnumbered, but newly forced tiles are numbered).\nThe configuration in Figure 5.3, referred to as PP, often appears in the case analysis; the two\nadjoining copies of P in the same orientation force a contradiction because nothing fits at the\nmarked point. Subsequently, when identifying forced tiles, as well as considering a tile as forced\nwhen it is the only one that would fit in a given place consistent with the matching conditions, we\nalso consider a tile as forced when the only alternative consistent with the matching conditions\nwould be to place a P tile in a way that produces this PP configuration.\n5.1. Cases involving T\nThe two A− edges of T must be adjacent to the A+ edge of H, while the B+ edge of T may be\nadjacent to either of the B− edges of H. Thus we have two cases for the configuration around\na T tile, which we refer to as T1 and T2 (Figure 5.4). As explained in the captions to this\nand subsequent figures, a sequence of deductions shows that any T in a tiling must occur in\ncase T1PF (Figure 5.9).","metadata":{"page":33,"title":"","author":""}},{"doc_id":"a8d0a02f-8b9a-4002-ac37-c2f8f0ea9126","text":"38\nDavid Smith et al.\n5.2. Cases with H not adjacent to T\nAny H not adjacent to a T tile must have a P tile adjacent to its A+ edge, while the B− edges\nmay each be adjacent to P or F. This results in four cases, which we call HPP (Figure 5.11),\nHPF (Figure 5.12), HFP (Figure 5.13), and HFF (Figure 5.14), and we proceed to draw\nfurther forced tiles in each of those cases, with consequences explained in the captions to those\nfigures.\n1\n2\n3\nFigure 5.11: Case HPP. Bisecting the P tiles and removing the forced F tiles produces the\nconfiguration of Figure 5.15, which we call T ′ and which combinatorially acts like T (with the\nedge segments indicated marked for matching conditions) in a tiling with the other supertiles.\nAlthough the forced F tiles are not included in T ′, the fact that they are forced will be used in\nthe proof that the supertiles must follow the matching conditions where they are adjacent to each\nother.","metadata":{"page":38,"title":"","author":""}},{"doc_id":"a8d0a02f-8b9a-4002-ac37-c2f8f0ea9126","text":"40\nDavid Smith et al.\n1\n2\n3\n4\n5\n6\nFigure 5.14: Case HFF. Bisecting the P and F tiles produces the configuration of Figure 5.18,\nwhich we call F ′ and which combinatorially acts like F (with the edge segments indicated\nmarked for matching conditions) in a tiling with the other supertiles.\n5.3. The supertiles\nA−\n2\nA−\n2\nB+\n2\nT\nFigure 5.15: Supertile T ′, alongside corresponding T\nThe previous arguments have shown that every H or T tile appears in a configuration corre-\nsponding to the supertiles T ′, H′, P ′ or T ′. We now provide more detailed rules for allocating\neach H or T tile, and, after bisecting all P and F tiles, each half of such a tile, to groupings\nof tiles, such that each tile is allocated to exactly one grouping, the groupings all have the form\nof one of the supertiles, all symmetries of the original tiling are also symmetries of the tiling\nby supertiles (this property follows immediately from the form of the rules, which do not in-\nvolve any arbitrary choices that could break symmetry), and the supertiles adjoin each other in\naccordance with the matching conditions indicated (A+\n2 adjoining A−\n2 , B+\n2 adjoining B−\n2 , X+\n2\nadjoining X−\n2 , F +\n2 adjoining F −\n2 , and L2 adjoining L2).\n• Each T tile is allocated to a H′ supertile, along with all the H tiles adjacent to that T.\n• Each H tile in case HPP is allocated to a T ′ supertile.\n• Each H tile in case HFP is allocated to a P ′ supertile, along with the H tile in case HPF\nshown in Figure 5.13.","metadata":{"page":40,"title":"","author":""}},{"doc_id":"a8d0a02f-8b9a-4002-ac37-c2f8f0ea9126","text":"preprint\n19\nFigure 2.12: An excerpt from a very large patch generated using the substitution system presented\nin this section. In the right half of the drawing, hats belonging to F metatiles are coloured black,\nto highlight the interlocking tree structures formed by the triskelions and the other metatiles.\nThe ideas presented in this section are sufficient to show that the hat does in fact tile the\nplane. Figure 2.12 offers a final large patch of tiles as a demonstration. On the right side of the\nillustration we observe that the tiles belonging to triskelions form a connected tree structure that\ninterlocks with a tree formed from the remaining tiles. This structure is reminiscent of those\nfound in other aperiodic tilings, such as the Taylor-Socolar tiling and the 1 + ϵ + ϵ2 tiling.\nHowever, exhibiting a tiling is usually the easy part of a proof of aperiodicity; it is also\nnecessary to prove that none of the tilings admitted by the hat can be periodic. In the next\nsection we present a novel geometric proof of aperiodicity. Then, in Sections 4 and 5 we turn\nto a more standard combinatorial argument that the matching rules implied by the substitution\nsystem shown earlier are forced in tilings by the hat.\n3. Aperiodicity via coupling of polyiamond tilings\nIn this section, we prove the following result:\nTheorem 3.1. Let T be a tiling by the hat polykite. Then T is not strongly periodic.\nAs noted in Section 1.3, a tile that does not admit strongly periodic tilings also cannot admit\nweakly periodic tilings. Therefore, together with the substitution system outlined in Section 2,","metadata":{"page":19,"title":"","author":""}},{"doc_id":"a8d0a02f-8b9a-4002-ac37-c2f8f0ea9126","text":"preprint\n39\n1\n2\n3\n4\nFigure 5.12: Case HPF. The second H cannot be adjacent to a T; it must thus itself be in case\nHFP or HFF, and each of those cases turns out to force the H tile in case HPF.\n1\n2\n3\n4\n5\nFigure 5.13: Case HFP. Bisecting the P and F tiles produces the configuration of Figure 5.17,\nwhich we call P ′ and which combinatorially acts like P (with the edge segments indicated\nmarked for matching conditions) in a tiling with the other supertiles.","metadata":{"page":39,"title":"","author":""}},{"doc_id":"a8d0a02f-8b9a-4002-ac37-c2f8f0ea9126","text":"preprint\n15\nFigure 2.7: The first four iterations of the H metatile and its supertiles. At each level, tiles\npartially overlap the boundary of their supertile. Overlaps are acceptable here, because the su-\npertile will be met by neighbouring supertiles with the same configuration of smaller tiles on its\nboundary.\nThe shapes of each generation of supertiles are different from those of the generation before\nit. However, by normalizing the tiles for size, we have computed that they quickly converge on\na fixed point, a set of tiles that truly do yield scaled copies of themselves under the construc-\ntion in Figure 2.6. These converged tile shapes are particularly interesting because they can be\nused to define a geometric substitution system that operates via inflation and replacement. The\nconverged tiles, together with their substitution rules, are shown in Figure 2.8. By virtue of its\nconnection to the original metatiles in Figure 2.5, we know that this substitution tiling is ape-\nriodic when the tiles are endowed with suitable matching conditions on their edges. We can\nalso use this system as an alternative means of constructing patches of hats. We cannot sim-\nply associate a cluster of hats rigidly with each converged tile, but a patch of converged tiles is\ncombinatorially equivalent to a corresponding patch of metatiles, which are equipped with hats.\nIf we rescale the converged tiles so that the short H edges have unit length, then all tile edges\nexcept the two F edges adjacent to a triskelion centre will have lengths in Z[ϕ], where ϕ is the\ngolden ratio. Furthermore, this substitution system has an inflation factor of ϕ2. The factor of ϕ2\ncan also be derived algebraically, through an eigenvalue computation on the substitution matrix","metadata":{"page":15,"title":"","author":""}}],"total_pages":89}Alright, let's break this down like you're five years old!

Imagine you have a big box of LEGO blocks. Some of these blocks are small, and some are big. The small blocks are like "tiles," and the big blocks made up of several small blocks stuck together are like "metatiles."

Now, when you play with these blocks, you notice that some blocks fit together in a special way. This is kind of like a puzzle. When you put these blocks together in the right way, you can make bigger and bigger shapes. These bigger shapes are like "supertiles."

In the paper you shared, they're talking about how to put these blocks (tiles and metatiles) together in the right way to make cool patterns and shapes. They even have special names for these blocks and the ways they fit together.

Here's a simple breakdown of what the paper says:

1. Metatiles and Tiles: Just like our LEGO blocks, they have small pieces (tiles) and bigger pieces made of small pieces (metatiles). They're trying to figure out how these pieces fit together.

2. Bisection of Tiles: Imagine you have a LEGO block, and you cut it in half. That's what "bisection" means. They're doing this to see how the pieces fit together in different ways.

3. Supertiles: When you put many LEGO blocks together in a special pattern, you get a big shape. This big shape is like a "supertile." The paper talks about how these supertiles are made and how they fit with each other.

4. Matching Rules: Just like in a puzzle, some pieces fit together, and some don't. The paper talks about the rules for how these pieces (tiles and metatiles) fit together.

5. Pictures and Diagrams: The paper has many pictures showing how these tiles, metatiles, and supertiles look and fit together.

I hope that helps! If you have specific questions or want to know more about a particular part, let me know!Explain it to me so that I can implement a drawing algorithm using html5 canvasAlright! Let's dive into how you can implement a drawing algorithm for tiles and metatiles using the HTML5 canvas. I'll provide a high-level overview and then break it down into steps:

High-Level Overview:

1. Canvas Setup: Initialize an HTML5 canvas where you'll draw the tiles and metatiles.
2. Tile Drawing Function: Create a function to draw individual tiles based on their type and position.
3. Metatile Drawing Function: Create a function to draw metatiles, which are essentially groups of tiles.
4. User Interaction: Allow users to place tiles and metatiles on the canvas and see how they fit together.

Step-by-Step Implementation:

1. Canvas Setup:

First, set up your HTML and canvas element:

<!DOCTYPE html>
<html lang="en">
<head>
    <meta charset="UTF-8">
    <meta name="viewport" content="width=device-width, initial-scale=1.0">
    <title>Tile Drawer</title>
</head>
<body>
    <canvas id="tileCanvas" width="800" height="600"></canvas>
    <script src="script.js"></script>
</body>
</html>

2. Tile Drawing Function:

In your script.js, start by getting a reference to the canvas and its 2D context:

const canvas = document.getElementById('tileCanvas');
const ctx = canvas.getContext('2d');

Now, create a function to draw individual tiles:

function drawTile(x, y, type) {
    // Depending on the type, draw the tile
    if (type === 'square') {
        ctx.fillStyle = 'blue';
        ctx.fillRect(x, y, 50, 50);
    }
    // Add more tile types as needed
}

3. Metatile Drawing Function:

A metatile is a group of tiles. Let's say a metatile is a group of four square tiles:

function drawMetatile(x, y) {
    drawTile(x, y, 'square');
    drawTile(x + 50, y, 'square');
    drawTile(x, y + 50, 'square');
    drawTile(x + 50, y + 50, 'square');
}

4. User Interaction:

Allow users to click on the canvas to draw tiles or metatiles:

canvas.addEventListener('click', function(event) {
    const x = event.clientX - canvas.offsetLeft;
    const y = event.clientY - canvas.offsetTop;

    // For simplicity, let's draw a metatile on click
    drawMetatile(x, y);
});

This is a basic setup. Depending on the complexity of your tiles, metatiles, and the rules from the scientific paper, you'll need to expand and adjust the drawing functions. You can also add features like zooming, panning, or a UI to select different tile types.The pdf has a lot of detail about how to draw a tile and then how to transform that tile into the gamma tile, etc. Please be more detailed{ "doc_id": "a8d0a02f-8b9a-4002-ac37-c2f8f0ea9126", "query": "drawing a tile" }{"results":[{"doc_id":"a8d0a02f-8b9a-4002-ac37-c2f8f0ea9126","text":"34\nDavid Smith et al.\n1\n2\nFigure 5.3: A common impossible configuration, referred to as PP\nCase T1\nCase T2\nFigure 5.4: Cases T1 and T2. Consider the three marked places in each of T1 and T2. These can\nbe filled with either P or H. On a side of the figure where there are two B− edges, the marked\nplace cannot be filled with H, because that would result in a 60◦ angle between two B− edges,\nwhich cannot be filled. So both those sides must have P in the marked place, while the third\nside may have H (oriented to avoid such a 60◦ angle between two B− edges; subsequently, when\nthe same situation arises, we just consider the orientation of the H to be forced without further\ncomment) or P. This results in four cases, which we call T1P (Figure 5.5), T2P (Figure 5.6),\nT1H (Figure 5.7) and T2H (Figure 5.8), and we proceed to draw further forced tiles in each of\nthose cases.","metadata":{"page":34,"title":"","author":""}},{"doc_id":"a8d0a02f-8b9a-4002-ac37-c2f8f0ea9126","text":"88\nDavid Smith et al.\n[GS09]\nChaim Goodman-Strauss. Regular production systems and triangle tilings. Theoret.\nComput. Sci., 410(16):1534–1549, 2009. doi:10.1016/j.tcs.2008.12.012.\n[GS16]\nBranko Gr¨unbaum and G.C. Shephard. Tilings and Patterns. Dover, second edition,\n2016.\n[GT21]\nRachel Greenfeld and Terence Tao. Undecidable translational tilings with only two\ntiles, or one nonabelian tile. 2021. arXiv:2108.07902.\n[GT22]\nRachel Greenfeld and Terence Tao. A counterexample to the periodic tiling conjec-\nture. 2022. arXiv:2211.15847.\n[Gum96] Petra Gummelt. Penrose tilings as coverings of congruent decagons. Geom. Dedicata,\n62(1):1–17, 1996. doi:10.1007/BF00239998.\n[Hee35]\nH. Heesch.\nAufbau der Ebene aus kongruenten Bereichen.\nNachr. Ges. Wiss.\nG¨ottingen, Math.-Phys. Kl. I, N. F., 1:115–117, 1935.\n[Hil02]\nDavid Hilbert. Mathematical problems. Bull. Amer. Math. Soc., 8(10):437–479, 1902.\ndoi:10.1090/S0002-9904-1902-00923-3.\n[JR21]\nEmmanuel Jeandel and Micha¨el Rao. An aperiodic set of 11 Wang tiles. Adv. Comb.,\n(1):1–37, 2021. doi:10.19086/aic.18614.\n[JS97]\nHyeong-Chai Jeong and Paul J. Steinhardt. Constructing Penrose-like tilings from a\nsingle prototile and the implications for quasicrystals. Phys. Rev. B, 55:3520–3532,\nFeb 1997. doi:10.1103/PhysRevB.55.3520.\n[Kap22]\nCraig S. Kaplan. Heesch numbers of unmarked polyforms. Contributions to Discrete\nMathematics, 17(2):150–171, 2022. URL: https://cdm.ucalgary.ca/article/\nview/72886.\n[Ken92]\nRichard Kenyon. Rigidity of planar tilings. Invent. Math., 107(3):637–651, 1992.\ndoi:10.1007/BF01231905.\n[Ken93]\nRichard Kenyon. Erratum: “Rigidity of planar tilings”. Invent. Math., 112(1):223,\n1993. doi:10.1007/BF01232432.\n[Ken96]\nRichard Kenyon. A group of paths in R2. Trans. Amer. Math. Soc., 348(8):3155–3172,\n1996. doi:10.1090/S0002-9947-96-01562-0.\n[Man04] Casey Mann.\nHeesch’s tiling problem.\nThe American Mathematical Monthly,\n111(6):509–517, 2004. doi:10.1080/00029890.2004.11920105.\n[MM98]\nG. A. Margulis and S. Mozes. Aperiodic tilings of the hyperbolic plane by convex\npolygons. Israel J. Math., 107:319–325, 1998. doi:10.1007/BF02764015.\n[Moz97] Shahar Mozes. Aperiodic tilings. Invent. Math., 128(3):603–611, 1997. doi:10.\n1007/s002220050153.\n[MT16]\nCasey Mann and B. Charles Thomas. Heesch numbers of edge-marked polyforms.\nExperimental Mathematics, 25(3):281–294, 2016. doi:10.1080/10586458.2015.\n1096867.","metadata":{"page":88,"title":"","author":""}},{"doc_id":"a8d0a02f-8b9a-4002-ac37-c2f8f0ea9126","text":"preprint\n77\n1-patch 13 (central tile class FP1)\n1-patch 14 (central tile class F2)\n1-patch 15 (eliminated by trying to\nsurround shaded tile)\n1-patch 16 (eliminated by trying to\nsurround shaded tile)\n1-patch 17 (eliminated by trying to\nsurround shaded tile)\n1-patch 18 (eliminated by trying to\nsurround shaded tile)\nFigure B.3: 1-patches (part 3)","metadata":{"page":77,"title":"","author":""}},{"doc_id":"a8d0a02f-8b9a-4002-ac37-c2f8f0ea9126","text":"40\nDavid Smith et al.\n1\n2\n3\n4\n5\n6\nFigure 5.14: Case HFF. Bisecting the P and F tiles produces the configuration of Figure 5.18,\nwhich we call F ′ and which combinatorially acts like F (with the edge segments indicated\nmarked for matching conditions) in a tiling with the other supertiles.\n5.3. The supertiles\nA−\n2\nA−\n2\nB+\n2\nT\nFigure 5.15: Supertile T ′, alongside corresponding T\nThe previous arguments have shown that every H or T tile appears in a configuration corre-\nsponding to the supertiles T ′, H′, P ′ or T ′. We now provide more detailed rules for allocating\neach H or T tile, and, after bisecting all P and F tiles, each half of such a tile, to groupings\nof tiles, such that each tile is allocated to exactly one grouping, the groupings all have the form\nof one of the supertiles, all symmetries of the original tiling are also symmetries of the tiling\nby supertiles (this property follows immediately from the form of the rules, which do not in-\nvolve any arbitrary choices that could break symmetry), and the supertiles adjoin each other in\naccordance with the matching conditions indicated (A+\n2 adjoining A−\n2 , B+\n2 adjoining B−\n2 , X+\n2\nadjoining X−\n2 , F +\n2 adjoining F −\n2 , and L2 adjoining L2).\n• Each T tile is allocated to a H′ supertile, along with all the H tiles adjacent to that T.\n• Each H tile in case HPP is allocated to a T ′ supertile.\n• Each H tile in case HFP is allocated to a P ′ supertile, along with the H tile in case HPF\nshown in Figure 5.13.","metadata":{"page":40,"title":"","author":""}},{"doc_id":"a8d0a02f-8b9a-4002-ac37-c2f8f0ea9126","text":"preprint\n39\n1\n2\n3\n4\nFigure 5.12: Case HPF. The second H cannot be adjacent to a T; it must thus itself be in case\nHFP or HFF, and each of those cases turns out to force the H tile in case HPF.\n1\n2\n3\n4\n5\nFigure 5.13: Case HFP. Bisecting the P and F tiles produces the configuration of Figure 5.17,\nwhich we call P ′ and which combinatorially acts like P (with the edge segments indicated\nmarked for matching conditions) in a tiling with the other supertiles.","metadata":{"page":39,"title":"","author":""}},{"doc_id":"a8d0a02f-8b9a-4002-ac37-c2f8f0ea9126","text":"38\nDavid Smith et al.\n5.2. Cases with H not adjacent to T\nAny H not adjacent to a T tile must have a P tile adjacent to its A+ edge, while the B− edges\nmay each be adjacent to P or F. This results in four cases, which we call HPP (Figure 5.11),\nHPF (Figure 5.12), HFP (Figure 5.13), and HFF (Figure 5.14), and we proceed to draw\nfurther forced tiles in each of those cases, with consequences explained in the captions to those\nfigures.\n1\n2\n3\nFigure 5.11: Case HPP. Bisecting the P tiles and removing the forced F tiles produces the\nconfiguration of Figure 5.15, which we call T ′ and which combinatorially acts like T (with the\nedge segments indicated marked for matching conditions) in a tiling with the other supertiles.\nAlthough the forced F tiles are not included in T ′, the fact that they are forced will be used in\nthe proof that the supertiles must follow the matching conditions where they are adjacent to each\nother.","metadata":{"page":38,"title":"","author":""}}],"total_pages":89}Alright, let's delve deeper into the details provided in the paper and translate them into actionable steps for your drawing algorithm using the HTML5 canvas.

1. Tile Drawing:

The paper mentions different types of tiles and their configurations. For instance, there are references to tiles like H, P, F, and T. Each of these tiles has a specific shape and orientation.

Actionable Steps:

• Define each tile type as a function that draws its shape on the canvas. For example, if H is a square, your function might look like:

  function drawH(x, y) {
      ctx.fillStyle = 'blue';
      ctx.fillRect(x, y, 50, 50);
  }
  

2. Bisecting Tiles:

The paper talks about bisecting certain tiles, like P and F. This means you'll be dividing these tiles into two parts.

Actionable Steps:

• Create a function that draws the bisected version of a tile. This could involve drawing a line through the tile or changing its color to indicate it's been bisected.

3. Metatiles and Supertiles:

The paper describes how tiles fit together to form larger structures called metatiles or supertiles. For example, T' is a supertile formed from other tiles.

Actionable Steps:

• Define functions to draw metatiles or supertiles. These functions will call the individual tile drawing functions in a specific arrangement. For instance:

  function drawTPrime(x, y) {
      drawH(x, y);
      drawP(x + 50, y);
      // ... and so on for other tiles that make up T'
  }
  

4. Matching Conditions:

The paper emphasizes how tiles fit together based on matching conditions. This is crucial for ensuring that the tiles and metatiles form a coherent pattern.

Actionable Steps:

• Implement checks in your drawing functions to ensure tiles are placed according to these matching conditions. For instance, if a P tile can only be adjacent to an H tile on one side, check the neighboring positions before drawing.

5. User Interaction and Visualization:

To make this interactive, allow users to select a tile type and place it on the canvas. Then, as they place tiles, ensure the matching conditions are met.

Actionable Steps:

• Add event listeners to the canvas to detect clicks.
• Depending on the selected tile type, draw the tile at the clicked position.
• Before drawing, check the matching conditions to ensure the tile can be placed there.

6. Advanced Features:

The paper has diagrams and figures that show specific configurations of tiles and metatiles. You can implement a feature where users can select these configurations and visualize them on the canvas.

Actionable Steps:

• Create predefined configurations based on the figures in the paper.
• Allow users to select and visualize these configurations on the canvas.

Remember, the exact shapes, orientations, and matching conditions of the tiles will depend on the detailed descriptions and diagrams in the paper. The steps provided here are a general guideline. You'll need to adapt and expand them based on the specific details from the paper.If you read the paper, you know that H is not a square. Can you try again?{ "doc_id": "a8d0a02f-8b9a-4002-ac37-c2f8f0ea9126", "query": "description of H tile" }{"results":[{"doc_id":"a8d0a02f-8b9a-4002-ac37-c2f8f0ea9126","text":"40\nDavid Smith et al.\n1\n2\n3\n4\n5\n6\nFigure 5.14: Case HFF. Bisecting the P and F tiles produces the configuration of Figure 5.18,\nwhich we call F ′ and which combinatorially acts like F (with the edge segments indicated\nmarked for matching conditions) in a tiling with the other supertiles.\n5.3. The supertiles\nA−\n2\nA−\n2\nB+\n2\nT\nFigure 5.15: Supertile T ′, alongside corresponding T\nThe previous arguments have shown that every H or T tile appears in a configuration corre-\nsponding to the supertiles T ′, H′, P ′ or T ′. We now provide more detailed rules for allocating\neach H or T tile, and, after bisecting all P and F tiles, each half of such a tile, to groupings\nof tiles, such that each tile is allocated to exactly one grouping, the groupings all have the form\nof one of the supertiles, all symmetries of the original tiling are also symmetries of the tiling\nby supertiles (this property follows immediately from the form of the rules, which do not in-\nvolve any arbitrary choices that could break symmetry), and the supertiles adjoin each other in\naccordance with the matching conditions indicated (A+\n2 adjoining A−\n2 , B+\n2 adjoining B−\n2 , X+\n2\nadjoining X−\n2 , F +\n2 adjoining F −\n2 , and L2 adjoining L2).\n• Each T tile is allocated to a H′ supertile, along with all the H tiles adjacent to that T.\n• Each H tile in case HPP is allocated to a T ′ supertile.\n• Each H tile in case HFP is allocated to a P ′ supertile, along with the H tile in case HPF\nshown in Figure 5.13.","metadata":{"page":40,"title":"","author":""}},{"doc_id":"a8d0a02f-8b9a-4002-ac37-c2f8f0ea9126","text":"preprint\n37\n1\nFigure 5.9: Case T1PF. Any T in a tiling must occur in this case. Bisecting the P and F tiles in\nthat case produces the configuration of Figure 5.16, which we call H′ and which combinatorially\nacts like H (with the edge segments indicated marked for matching conditions) in a tiling along\nwith configurations T ′, P ′, and F ′.\n1\n2\n3\n4\n5\n6\n7\nFigure 5.10: Case T1PP, eliminated because PP occurs at the marked point.","metadata":{"page":37,"title":"","author":""}},{"doc_id":"a8d0a02f-8b9a-4002-ac37-c2f8f0ea9126","text":"76\nDavid Smith et al.\n1-patch 7 (central tile class H2)\n1-patch 8 (central tile class H2)\n1-patch 9 (central tile class H1)\n1-patch 10 (central tile class H4)\n1-patch 11 (eliminated by trying to\nsurround shaded tile)\n1-patch 12 (central tile class FP1)\nFigure B.3: 1-patches (part 2)","metadata":{"page":76,"title":"","author":""}},{"doc_id":"a8d0a02f-8b9a-4002-ac37-c2f8f0ea9126","text":"42\nDavid Smith et al.\n• Each H tile in case HFF is allocated to an F ′ supertile, along with the H tile in case HPF\nshown in Figure 5.14.\n• Each H tile in case HPF was allocated to a supertile by exactly one of the previous two\nrules.\n• Each half of a P tile, and the upper half of each F tile, is adjacent to exactly one H tile\nalong its A− or B+ edge, and is allocated to the same supertile as that H tile. (Where both\nhalves are allocated to the same supertile, the bisection isn’t shown in the diagrams of the\nsupertiles, to simplify those diagrams.)\n• It remains to allocate the lower halves of F tiles. Each such lower half has a X− edge\nbetween a L edge and a F + edge; it is allocated to the same supertile as the H tile adjacent\nto that X− edge. For this allocation rule to be well defined, we need to show that this X−\nedge is indeed adjacent to a H tile. The only other possibility not violating the matching\nrules would be adjacency to another F tile, as shown in Figure 5.19, but nothing fits at the\nmarked point without violating the matching rules, so that cannot occur.\nThe X− edge referenced in the last allocation rule cannot be adjacent to any of the exposed\nX+ edges of H tiles in supertiles T ′, P ′ or F ′ without violating the matching rules. Thus all\nlower halves of F tiles are the ones that appear on the diagrams of the supertiles, and we have\nshown that the tiling is partitioned into the supertiles.\nFigure 5.19: Impossible adjacency of two F tiles\nWe now show that the supertiles adjoin each other in accordance with the matching condi-\ntions indicated. First, we examine P + edges (appearing in A−\n2 and B−\n2 ) and P − edges (appearing\nin A+\n2 and B+\n2 ). B−\n2 and A+\n2 appear only in H′, where their P + and P − edges cannot meet with-\nout tiles intersecting. So B−\n2 can only join to B+\n2 and A+\n2 can only join to A−\n2 .\nNext we show that the converse holds: A−\n2 can only join to A+\n2 and B+\n2 can only join to B−\n2 .\nFor a contradiction, suppose that the P + and P − edges in some A−\n2 and B+\n2 are joined. If the\nB+\n2 comes from a P ′ supertile, then that P − bisects tile 5 in case HFP. Adjacent tiles 5 and 1 in\nthat configuration both have B+ edges, which must both be adjacent to H tiles; those H tiles are\nadjacent to each other, which can only occur in a supertile H′, which does not have an A−\n2 edge.\nThe same argument applies in the case of an F ′ supertile, considering tiles 6 and 2 in case HFF.\nIf the P + in an A−\n2 from P ′ is joined to a B+\n2 , a similar argument applies (considering tile 2 and\nan adjacent unnumbered tile in case HFP). So the only remaining case would be if both edges\ncome from supertile T ′, but that is inconsistent with the F tiles forced in case HPP.","metadata":{"page":42,"title":"","author":""}},{"doc_id":"a8d0a02f-8b9a-4002-ac37-c2f8f0ea9126","text":"38\nDavid Smith et al.\n5.2. Cases with H not adjacent to T\nAny H not adjacent to a T tile must have a P tile adjacent to its A+ edge, while the B− edges\nmay each be adjacent to P or F. This results in four cases, which we call HPP (Figure 5.11),\nHPF (Figure 5.12), HFP (Figure 5.13), and HFF (Figure 5.14), and we proceed to draw\nfurther forced tiles in each of those cases, with consequences explained in the captions to those\nfigures.\n1\n2\n3\nFigure 5.11: Case HPP. Bisecting the P tiles and removing the forced F tiles produces the\nconfiguration of Figure 5.15, which we call T ′ and which combinatorially acts like T (with the\nedge segments indicated marked for matching conditions) in a tiling with the other supertiles.\nAlthough the forced F tiles are not included in T ′, the fact that they are forced will be used in\nthe proof that the supertiles must follow the matching conditions where they are adjacent to each\nother.","metadata":{"page":38,"title":"","author":""}},{"doc_id":"a8d0a02f-8b9a-4002-ac37-c2f8f0ea9126","text":"preprint\n39\n1\n2\n3\n4\nFigure 5.12: Case HPF. The second H cannot be adjacent to a T; it must thus itself be in case\nHFP or HFF, and each of those cases turns out to force the H tile in case HPF.\n1\n2\n3\n4\n5\nFigure 5.13: Case HFP. Bisecting the P and F tiles produces the configuration of Figure 5.17,\nwhich we call P ′ and which combinatorially acts like P (with the edge segments indicated\nmarked for matching conditions) in a tiling with the other supertiles.","metadata":{"page":39,"title":"","author":""}}],"total_pages":89}I apologize for the oversight. Let's dive into the details of the H tile based on the information provided in the paper.

From the extracted details:

1. Cases with H not adjacent to T: The paper mentions that any H tile not adjacent to a T tile must have a P tile adjacent to its A+ edge. The B− edges may each be adjacent to P or F. This results in four cases: HPP, HPF, HFP, and HFF. Each of these cases has specific configurations and forced tiles.

2. Case HPP: In this case, bisecting the P tiles and removing the forced F tiles produces a configuration called T', which acts like T in a tiling with other supertiles.

3. Case HPF: The second H cannot be adjacent to a T. It must be in case HFP or HFF. Each of these cases forces the H tile in case HPF.

4. Case HFP: Bisecting the P and F tiles in this case produces a configuration called P', which acts like P in a tiling with other supertiles.

Based on these details, the H tile seems to have specific configurations and interactions with other tiles like P, F, and T. The exact shape and orientation of the H tile are not explicitly mentioned in the extracted details, but its interactions and configurations with other tiles are described.

To implement a drawing algorithm for the H tile and its configurations:

1. Define the H Tile: Create a function to draw the H tile on the canvas based on its described shape and orientation.

2. Handle Configurations: Create functions to handle the different configurations of the H tile (HPP, HPF, HFP). These functions should draw the H tile and its adjacent tiles based on the described configurations.

3. User Interaction: Allow users to select a configuration (e.g., HPP, HPF) and visualize it on the canvas. Ensure that the tiles are placed according to the described configurations and interactions.

4. Matching Conditions: Implement checks to ensure that the tiles are placed according to the described matching conditions. For instance, ensure that an H tile not adjacent to a T tile has a P tile adjacent to its A+ edge.

5. Bisection and Forced Tiles: Implement the bisection of tiles like P and F based on the described cases and handle the forced tiles accordingly.

Remember, the exact drawing and placement of the H tile will depend on its shape, orientation, and interactions with other tiles as described in the paper. The steps provided here are based on the extracted details and should be adapted based on the complete information from the paper.

Read Scientific Paper Help created at Tue Jan 20 1970 15:17:21 GMT+0000 (Coordinated Universal Time)