summaryrefslogtreecommitdiff
path: root/project
diff options
context:
space:
mode:
Diffstat (limited to 'project')
-rw-r--r--project/assets/css/accordian.css39
-rw-r--r--project/assets/css/numberwall.css38
-rw-r--r--project/assets/css/steelman.css39
-rw-r--r--project/assets/img/pagoda_wall_mod_5.pngbin0 -> 12030 bytes
-rw-r--r--project/complexity.yml1
-rw-r--r--project/context/articles.json6
-rw-r--r--project/templates/numberwall.xhtml1234
-rw-r--r--project/templates/steelman.xhtml1
8 files changed, 1319 insertions, 39 deletions
diff --git a/project/assets/css/accordian.css b/project/assets/css/accordian.css
new file mode 100644
index 0000000..9c52e5d
--- /dev/null
+++ b/project/assets/css/accordian.css
@@ -0,0 +1,39 @@
+
+
+div.accordian {
+ margin: 1em 2em 2em 2em;
+}
+
+
+.accordian > input[name="collapse"] {
+ display: none;
+}
+
+
+.accordian .hidden_content {
+ overflow: hidden;
+ height: 0;
+}
+
+
+.accordian label {
+ color: #ffffff;
+ cursor: pointer;
+ font-weight: normal;
+ padding: 0.5em;
+ background: #808080;
+}
+
+
+.accordian label:hover,
+.accordian label:focus {
+ color: #ffffff;
+ background: #000000;
+}
+
+
+.accordian > input[name="collapse"]:checked ~ .hidden_content {
+ height: auto;
+}
+
+
diff --git a/project/assets/css/numberwall.css b/project/assets/css/numberwall.css
new file mode 100644
index 0000000..8efcea8
--- /dev/null
+++ b/project/assets/css/numberwall.css
@@ -0,0 +1,38 @@
+
+
+table {
+ margin: 2em auto 2em auto;
+}
+
+
+
+
+table.rule {
+ border: 1px solid gray;
+ width: 90%;
+}
+
+table.rule th {
+ border: 1px solid gray;
+ width: 33%;
+}
+
+table.rule td {
+ padding: 0.5em 1em 0.5em 1em;
+ border: 1px solid gray;
+}
+
+
+
+
+table.wall {
+ border: none;
+ text-align: center;
+}
+
+table.wall td {
+ padding: 0.5em 0.4em 0.5em 0.4em;
+ border: none;
+}
+
+
diff --git a/project/assets/css/steelman.css b/project/assets/css/steelman.css
index 14bc84b..5b6b5c9 100644
--- a/project/assets/css/steelman.css
+++ b/project/assets/css/steelman.css
@@ -1,44 +1,5 @@
-div.accordian {
- margin: 1em 2em 2em 2em;
-}
-
-
-.accordian > input[name="collapse"] {
- display: none;
-}
-
-
-.accordian .hidden_content {
- overflow: hidden;
- height: 0;
-}
-
-
-.accordian label {
- color: #ffffff;
- cursor: pointer;
- font-weight: normal;
- padding: 0.5em;
- background: #808080;
-}
-
-
-.accordian label:hover,
-.accordian label:focus {
- color: #ffffff;
- background: #000000;
-}
-
-
-.accordian > input[name="collapse"]:checked ~ .hidden_content {
- height: auto;
-}
-
-
-
-
table {
margin: 2em auto 2em auto;
max-width: 60em;
diff --git a/project/assets/img/pagoda_wall_mod_5.png b/project/assets/img/pagoda_wall_mod_5.png
new file mode 100644
index 0000000..4634d6b
--- /dev/null
+++ b/project/assets/img/pagoda_wall_mod_5.png
Binary files differ
diff --git a/project/complexity.yml b/project/complexity.yml
index 58b56ca..0f14072 100644
--- a/project/complexity.yml
+++ b/project/complexity.yml
@@ -19,6 +19,7 @@ unexpanded_templates:
- "index.xhtml"
- "integral.xhtml"
- "links.xhtml"
+ - "numberwall.xhtml"
- "packrat.xhtml"
- "sokoban.xhtml"
- "steelman.xhtml"
diff --git a/project/context/articles.json b/project/context/articles.json
index fef380d..50611ab 100644
--- a/project/context/articles.json
+++ b/project/context/articles.json
@@ -1,6 +1,12 @@
[
{
+ "title": "Number Wall Visualiser",
+ "anchor": "/numberwall.xhtml",
+ "taglist": ["application", "mathematics", "programming"],
+ "postdate": "9/11/2022"
+ },
+ {
"title": "Contributions to Other Projects",
"anchor": "/contributions.xhtml",
"taglist": ["programming"],
diff --git a/project/templates/numberwall.xhtml b/project/templates/numberwall.xhtml
new file mode 100644
index 0000000..0e343ed
--- /dev/null
+++ b/project/templates/numberwall.xhtml
@@ -0,0 +1,1234 @@
+
+{%- extends "base_math.xhtml" -%}
+
+
+
+{%- block title -%}Number Wall Visualiser{%- endblock -%}
+
+
+
+{%- block footer -%}{{ math_footer ("numberwall.xhtml") }}{%- endblock -%}
+
+
+
+{%- block style %}
+ <link href="/css/numberwall.css" rel="stylesheet" />
+ <link href="/css/accordian.css" rel="stylesheet" />
+{% endblock -%}
+
+
+
+{%- block content %}
+<h4>Number Wall Visualiser</h4>
+
+<p>Git repository: <a href="/cgi-bin/cgit.cgi/number-wall">Link</a><br />
+Mathologer video:
+<a href="https://invidious.namazso.eu/watch?v=NO1_-qptr6c" class="external">Invidious</a>
+<a href="https://www.youtube.com/watch?v=NO1_-qptr6c" class="external">Youtube</a></p>
+
+<h5>9/11/2022</h5>
+
+
+<h5>Overview</h5>
+
+<p>On August 27, 2022, a youtube mathematician called the Mathologer published an interesting video
+on the mathematical constructs known as number walls. In the description of that video he also
+published a coding challenge of creating an implementation of the number wall algorithm.</p>
+
+<p>This is not an entry into that coding challenge. This is just a tribute.</p>
+
+<p>For various reasons this didn't end up finished in the time period required, and in any case it
+is not an online implementation and so doesn't qualify. But this project still has some fun tools to
+play around with for anyone with an interest in math. Amongst other things, elaborate pictures of
+the fractal square patterns produced by large number walls can be made.</p>
+
+<div class="figure">
+ <img src="/img/pagoda_wall_mod_5.png"
+ alt="Number wall of the Pagoda sequence modulo 5"
+ height="600"
+ width="600" />
+ <div class="figcaption">Number wall of the <a href="https://oeis.org/A301849" class="external">
+ Pagoda sequence</a> with cells equal to 0 mod 5 in orange and others in blue</div>
+</div>
+
+<p>It is strongly recommended that you watch the Mathologer's video for an explanation of what
+exactly a number wall is. If, for some reason, you don't want to do that or the video is
+inaccessible, then the rules of how to construct a number wall are detailed in the appendix.</p>
+
+
+<h5>Tools</h5>
+
+<p>There are three programs provided:</p>
+<ul>
+ <li><i>wallgen</i>, a command line program that generates a number wall from a given
+ input sequence in comma separated value format.</li>
+ <li><i>wallsolve</i>, a command line program that uses a number wall and polynomials constructed
+ from a given input sequence to find a recurrence relation that generates that sequence, and then
+ calculates the next integer using that recurrence relation.</li>
+ <li><i>visualwall</i>, a graphical program that creates a visual representation of a number wall
+ with cells equal to zero modulo a given number highlighted in one color, and all other cells
+ highlighted in a different color.</li>
+</ul>
+
+<p>An example of using <i>wallsolve</i> to calculate the next number in the sequence of integer
+cubes:</p>
+<div class="precontain">
+<pre>
+> wallsolve -i 1,8,27,64,125,216,343,512,729,1000
+S(n) = 4S(n-1) - 6S(n-2) + 4S(n-3) - S(n-4)
+Next: 1331
+>
+</pre>
+</div>
+
+<p>An example of <i>visualwall</i> has already been given in the overview in the form of the picture
+of the Pagoda sequence number wall, and the output of <i>wallgen</i> is too verbose to show here.
+</p>
+
+
+<h5>Implementation Details</h5>
+
+<p>All calculations to produce the number walls in these programs are done using the simple
+construction rules in the appendix. No determinants were used, as that method would be more
+computationally intensive.</p>
+
+<p>Since the methods of constructing a number wall given by the Mathologer assume a sequence that is
+infinite in both directions and any sequence used by these programs is finite, some compromises had
+to be made. In particular, <i>visualwall</i> uses a longer sequence to generate its number wall and
+then fits the largest square viewing area it can inside the resulting triangle of calculated values.
+</p>
+
+<p>The polynomial library made for <i>wallsolve</i> assumes that coefficients will always be
+integers. This applies even to the division implementation, with only integer division being used
+and the remainder term becoming larger than it may otherwise be expected to be if rationals were
+permitted. As a number wall calculated from an integer sequence (or a sequence of polynomials with
+integer coefficients) will always have integer values, this isn't actually a problem. But care had
+to be taken in number wall construction so that division operations were always left until last.</p>
+
+<p>The modulus values allowed by <i>visualwall</i> are not limited to prime numbers. If a composite
+modulus is selected then the highlighted portions of cells equal to zero modulo that number may not
+produce squares. The resulting pictures are, however, still interesting to examine.</p>
+
+<p>At present <i>visualwall</i> only has the option to display either the Pagoda sequence or a
+sequence of random numbers, unless a comma separated value formatted sequence is supplied in an
+external file. It would not be hard for anyone with a passing knowledge of OCaml to add in different
+sequences, but at this stage there is no need.</p>
+
+
+<h5>Appendix: How to Construct a Number Wall</h5>
+
+<p>A number wall is a grid of numbers that extends infinitely left, right, and down. It begins with
+an infinite row of ones at the top, then below that a row containing the input sequence. All other
+cells are computed according to the following rules.</p>
+
+<div class="accordian">
+ <input type="checkbox" name="collapse" id="handle1" />
+ <label for="handle1">Toggle Cross Rule</label>
+ <div class="hidden_content">
+
+<table class="rule">
+ <tr>
+ <th>Pattern</th>
+ <th>Relation</th>
+ <th>Notes</th>
+ </tr>
+ <tr>
+ <td>
+ <table class="wall">
+ <tr>
+ <td>&nbsp;</td>
+ <td>
+ <math xmlns="http://www.w3.org/1998/Math/MathML">
+ <mi>a</mi>
+ </math>
+ </td>
+ <td>&nbsp;</td>
+ </tr>
+ <tr>
+ <td>
+ <math xmlns="http://www.w3.org/1998/Math/MathML">
+ <mi>b</mi>
+ </math>
+ </td>
+ <td>
+ <math xmlns="http://www.w3.org/1998/Math/MathML">
+ <mi>c</mi>
+ </math>
+ </td>
+ <td>
+ <math xmlns="http://www.w3.org/1998/Math/MathML">
+ <mi>d</mi>
+ </math>
+ </td>
+ </tr>
+ <tr>
+ <td>&nbsp;</td>
+ <td>
+ <math xmlns="http://www.w3.org/1998/Math/MathML">
+ <mi>e</mi>
+ </math>
+ </td>
+ <td>&nbsp;</td>
+ </tr>
+ </table>
+ </td>
+ <td>
+ <math xmlns="http://www.w3.org/1998/Math/MathML">
+ <mrow>
+ <mi>a</mi>
+ <mo>&it;</mo>
+ <mi>e</mi>
+ <mo>+</mo>
+ <mi>b</mi>
+ <mo>&it;</mo>
+ <mi>d</mi>
+ <mo>=</mo>
+ <msup>
+ <mi>c</mi>
+ <mn>2</mn>
+ </msup>
+ </mrow>
+ </math>
+ <br />
+ <br />
+ or alternatively
+ <br />
+ <br />
+ <math xmlns="http://www.w3.org/1998/Math/MathML">
+ <mrow>
+ <mi>e</mi>
+ <mo>=</mo>
+ <mfrac>
+ <mrow>
+ <msup>
+ <mi>c</mi>
+ <mn>2</mn>
+ </msup>
+ <mo>-</mo>
+ <mi>b</mi>
+ <mo>&it;</mo>
+ <mi>d</mi>
+ </mrow>
+ <mi>a</mi>
+ </mfrac>
+ </mrow>
+ </math>
+ <br />
+ <br />
+ where
+ <math xmlns="http://www.w3.org/1998/Math/MathML">
+ <mrow>
+ <mi>a</mi>
+ <mo>&ne;</mo>
+ <mn>0</mn>
+ </mrow>
+ </math>
+ </td>
+ <td>
+ This rule is used to calculate the vast majority of cells.
+ </td>
+ </tr>
+</table>
+
+ </div>
+</div>
+
+<div class="accordian">
+ <input type="checkbox" name="collapse" id="handle2" />
+ <label for="handle2">Toggle Long Cross Rule</label>
+ <div class="hidden_content">
+
+<table class="rule">
+ <tr>
+ <th>Pattern</th>
+ <th>Relation</th>
+ <th>Notes</th>
+ </tr>
+ <tr>
+ <td>
+ <table class="wall">
+ <tr>
+ <td>&nbsp;</td>
+ <td>&nbsp;</td>
+ <td>
+ <math xmlns="http://www.w3.org/1998/Math/MathML">
+ <mi>a</mi>
+ </math>
+ </td>
+ <td>&nbsp;</td>
+ <td>&nbsp;</td>
+ </tr>
+ <tr>
+ <td>&nbsp;</td>
+ <td>&nbsp;</td>
+ <td>
+ <math xmlns="http://www.w3.org/1998/Math/MathML">
+ <mi>b</mi>
+ </math>
+ </td>
+ <td>&nbsp;</td>
+ <td>&nbsp;</td>
+ </tr>
+ <tr>
+ <td>
+ <math xmlns="http://www.w3.org/1998/Math/MathML">
+ <mi>c</mi>
+ </math>
+ </td>
+ <td>
+ <math xmlns="http://www.w3.org/1998/Math/MathML">
+ <mi>d</mi>
+ </math>
+ </td>
+ <td>&nbsp;</td>
+ <td>
+ <math xmlns="http://www.w3.org/1998/Math/MathML">
+ <mi>e</mi>
+ </math>
+ </td>
+ <td>
+ <math xmlns="http://www.w3.org/1998/Math/MathML">
+ <mi>f</mi>
+ </math>
+ </td>
+ </tr>
+ <tr>
+ <td>&nbsp;</td>
+ <td>&nbsp;</td>
+ <td>
+ <math xmlns="http://www.w3.org/1998/Math/MathML">
+ <mi>g</mi>
+ </math>
+ </td>
+ <td>&nbsp;</td>
+ <td>&nbsp;</td>
+ </tr>
+ <tr>
+ <td>&nbsp;</td>
+ <td>&nbsp;</td>
+ <td>
+ <math xmlns="http://www.w3.org/1998/Math/MathML">
+ <mi>h</mi>
+ </math>
+ </td>
+ <td>&nbsp;</td>
+ <td>&nbsp;</td>
+ </tr>
+ </table>
+ </td>
+ <td>
+ <math xmlns="http://www.w3.org/1998/Math/MathML">
+ <mrow>
+ <msup>
+ <mi>b</mi>
+ <mn>2</mn>
+ </msup>
+ <mo>&it;</mo>
+ <mi>h</mi>
+ <mo>+</mo>
+ <mi>a</mi>
+ <mo>&it;</mo>
+ <msup>
+ <mi>g</mi>
+ <mn>2</mn>
+ </msup>
+ <mo>=</mo>
+ <msup>
+ <mi>d</mi>
+ <mn>2</mn>
+ </msup>
+ <mo>&it;</mo>
+ <mi>f</mi>
+ <mo>+</mo>
+ <mi>c</mi>
+ <mo>&it;</mo>
+ <msup>
+ <mi>e</mi>
+ <mn>2</mn>
+ </msup>
+ </mrow>
+ </math>
+ <br />
+ <br />
+ or alternatively
+ <br />
+ <br />
+ <math xmlns="http://www.w3.org/1998/Math/MathML">
+ <mrow>
+ <mi>h</mi>
+ <mo>=</mo>
+ <mfrac>
+ <mrow>
+ <msup>
+ <mi>d</mi>
+ <mn>2</mn>
+ </msup>
+ <mo>&it;</mo>
+ <mi>f</mi>
+ <mo>+</mo>
+ <mi>c</mi>
+ <mo>&it;</mo>
+ <msup>
+ <mi>e</mi>
+ <mn>2</mn>
+ </msup>
+ <mo>-</mo>
+ <mi>a</mi>
+ <mo>&it;</mo>
+ <msup>
+ <mi>g</mi>
+ <mn>2</mn>
+ </msup>
+ </mrow>
+ <msup>
+ <mi>b</mi>
+ <mn>2</mn>
+ </msup>
+ </mfrac>
+ </mrow>
+ </math>
+ <br />
+ <br />
+ where
+ <math xmlns="http://www.w3.org/1998/Math/MathML">
+ <mi>b</mi>
+ <mo>&ne;</mo>
+ <mn>0</mn>
+ </math>
+ </td>
+ <td>
+ If cell
+ <math xmlns="http://www.w3.org/1998/Math/MathML">
+ <mi>a</mi>
+ </math>
+ goes out of bounds of the wall, it is considered equal to zero.
+ </td>
+ </tr>
+</table>
+
+ </div>
+</div>
+
+<div class="accordian">
+ <input type="checkbox" name="collapse" id="handle3" />
+ <label for="handle3">Toggle Horseshoe Rule</label>
+ <div class="hidden_content">
+
+<table class="rule">
+ <tr>
+ <th>Pattern</th>
+ <th>Relation</th>
+ <th>Notes</th>
+ </tr>
+ <tr>
+ <td>
+ <table class="wall">
+ <tr>
+ <td>
+ <math xmlns="http://www.w3.org/1998/Math/MathML">
+ <mi>a</mi>
+ </math>
+ </td>
+ <td>
+ <math xmlns="http://www.w3.org/1998/Math/MathML">
+ <mi>b</mi>
+ </math>
+ </td>
+ <td>
+ <math xmlns="http://www.w3.org/1998/Math/MathML">
+ <mi>c</mi>
+ </math>
+ </td>
+ <td>
+ <math xmlns="http://www.w3.org/1998/Math/MathML">
+ <mi>d</mi>
+ </math>
+ </td>
+ <td>
+ <math xmlns="http://www.w3.org/1998/Math/MathML">
+ <mi>e</mi>
+ </math>
+ </td>
+ <td>
+ <math xmlns="http://www.w3.org/1998/Math/MathML">
+ <mi>f</mi>
+ </math>
+ </td>
+ </tr>
+ <tr>
+ <td>
+ <math xmlns="http://www.w3.org/1998/Math/MathML">
+ <mi>g</mi>
+ </math>
+ </td>
+ <td>0</td>
+ <td>0</td>
+ <td>0</td>
+ <td>0</td>
+ <td>
+ <math xmlns="http://www.w3.org/1998/Math/MathML">
+ <mi>h</mi>
+ </math>
+ </td>
+ </tr>
+ <tr>
+ <td>
+ <math xmlns="http://www.w3.org/1998/Math/MathML">
+ <mi>i</mi>
+ </math>
+ </td>
+ <td>0</td>
+ <td>0</td>
+ <td>0</td>
+ <td>0</td>
+ <td>
+ <math xmlns="http://www.w3.org/1998/Math/MathML">
+ <mi>j</mi>
+ </math>
+ </td>
+ </tr>
+ <tr>
+ <td>
+ <math xmlns="http://www.w3.org/1998/Math/MathML">
+ <mi>k</mi>
+ </math>
+ </td>
+ <td>0</td>
+ <td>0</td>
+ <td>0</td>
+ <td>0</td>
+ <td>
+ <math xmlns="http://www.w3.org/1998/Math/MathML">
+ <mi>l</mi>
+ </math>
+ </td>
+ </tr>
+ <tr>
+ <td>
+ <math xmlns="http://www.w3.org/1998/Math/MathML">
+ <mi>m</mi>
+ </math>
+ </td>
+ <td>0</td>
+ <td>0</td>
+ <td>0</td>
+ <td>0</td>
+ <td>
+ <math xmlns="http://www.w3.org/1998/Math/MathML">
+ <mi>n</mi>
+ </math>
+ </td>
+ </tr>
+ <tr>
+ <td>
+ <math xmlns="http://www.w3.org/1998/Math/MathML">
+ <mi>o</mi>
+ </math>
+ </td>
+ <td>
+ <math xmlns="http://www.w3.org/1998/Math/MathML">
+ <mi>p</mi>
+ </math>
+ </td>
+ <td>
+ <math xmlns="http://www.w3.org/1998/Math/MathML">
+ <mi>q</mi>
+ </math>
+ </td>
+ <td>
+ <math xmlns="http://www.w3.org/1998/Math/MathML">
+ <mi>r</mi>
+ </math>
+ </td>
+ <td>
+ <math xmlns="http://www.w3.org/1998/Math/MathML">
+ <mi>s</mi>
+ </math>
+ </td>
+ <td>
+ <math xmlns="http://www.w3.org/1998/Math/MathML">
+ <mi>t</mi>
+ </math>
+ </td>
+ </tr>
+ </table>
+ </td>
+ <td>
+ <math xmlns="http://www.w3.org/1998/Math/MathML">
+ <mrow>
+ <mfrac>
+ <mrow>
+ <mi>W</mi>
+ <mo>&it;</mo>
+ <mi>X</mi>
+ </mrow>
+ <mrow>
+ <mi>Y</mi>
+ <mo>&it;</mo>
+ <mi>Z</mi>
+ </mrow>
+ </mfrac>
+ <mo>=</mo>
+ <mi>S</mi>
+ </mrow>
+ </math>
+ where
+ <br />
+ <br />
+ <math xmlns="http://www.w3.org/1998/Math/MathML">
+ <mi>W</mi>
+ </math>
+ is the common ratio of the arithmetic sequence
+ <math xmlns="http://www.w3.org/1998/Math/MathML">
+ <mrow>
+ <mi>a</mi>
+ <mo>,</mo>
+ <mi>b</mi>
+ <mo>,</mo>
+ <mi>c</mi>
+ <mo>,</mo>
+ <mi>d</mi>
+ <mo>,</mo>
+ <mi>e</mi>
+ <mo>,</mo>
+ <mi>f</mi>
+ </mrow>
+ </math>
+ <br />
+ <br />
+ <math xmlns="http://www.w3.org/1998/Math/MathML">
+ <mi>X</mi>
+ </math>
+ is the common ratio of the arithmetic sequence
+ <math xmlns="http://www.w3.org/1998/Math/MathML">
+ <mrow>
+ <mi>t</mi>
+ <mo>,</mo>
+ <mi>s</mi>
+ <mo>,</mo>
+ <mi>r</mi>
+ <mo>,</mo>
+ <mi>q</mi>
+ <mo>,</mo>
+ <mi>p</mi>
+ <mo>,</mo>
+ <mi>o</mi>
+ </mrow>
+ </math>
+ <br />
+ <br />
+ <math xmlns="http://www.w3.org/1998/Math/MathML">
+ <mi>Y</mi>
+ </math>
+ is the common ratio of the arithmetic sequence
+ <math xmlns="http://www.w3.org/1998/Math/MathML">
+ <mrow>
+ <mi>a</mi>
+ <mo>,</mo>
+ <mi>g</mi>
+ <mo>,</mo>
+ <mi>i</mi>
+ <mo>,</mo>
+ <mi>k</mi>
+ <mo>,</mo>
+ <mi>m</mi>
+ <mo>,</mo>
+ <mi>o</mi>
+ </mrow>
+ </math>
+ <br />
+ <br />
+ <math xmlns="http://www.w3.org/1998/Math/MathML">
+ <mi>Z</mi>
+ </math>
+ is the common ratio of the arithmetic sequence
+ <math xmlns="http://www.w3.org/1998/Math/MathML">
+ <mrow>
+ <mi>t</mi>
+ <mo>,</mo>
+ <mi>n</mi>
+ <mo>,</mo>
+ <mi>l</mi>
+ <mo>,</mo>
+ <mi>j</mi>
+ <mo>,</mo>
+ <mi>h</mi>
+ <mo>,</mo>
+ <mi>f</mi>
+ </mrow>
+ </math>
+ <br />
+ <br />
+ <math xmlns="http://www.w3.org/1998/Math/MathML">
+ <mi>S</mi>
+ </math>
+ is equal to
+ <math xmlns="http://www.w3.org/1998/Math/MathML">
+ <mn>1</mn>
+ </math>
+ if the dimension of the zero square is even, or
+ <math xmlns="http://www.w3.org/1998/Math/MathML">
+ <mn>-1</mn>
+ </math>
+ if the dimension is odd
+ </td>
+ <td>
+ All cells surrounding a square of zeros are guaranteed to be nonzero.<br />
+ <br />
+ This rule is applicable to squares of zeros of any finite size, but for the sake of
+ simplicity the previous rule should be used for isolated zeros.<br />
+ <br />
+ The cells that this rule should be targeted at calculating are those on the bottom row
+ below the square of zeros. Reformulating the math to solve for those cells is left as an
+ exercise to the reader.
+ </td>
+ </tr>
+</table>
+
+ </div>
+</div>
+
+<div class="accordian">
+ <input type="checkbox" name="collapse" id="handle4" />
+ <label for="handle4">Toggle Broken Cross Rule</label>
+ <div class="hidden_content">
+
+<table class="rule">
+ <tr>
+ <th>Pattern</th>
+ <th>Relation</th>
+ <th>Notes</th>
+ </tr>
+ <tr>
+ <td>
+ <table class="wall">
+ <tr>
+ <td>&nbsp;</td>
+ <td>&nbsp;</td>
+ <td>
+ <math xmlns="http://www.w3.org/1998/Math/MathML">
+ <msub>
+ <mi>a</mi>
+ <mn>1</mn>
+ </msub>
+ </math>
+ </td>
+ <td>
+ <math xmlns="http://www.w3.org/1998/Math/MathML">
+ <msub>
+ <mi>a</mi>
+ <mn>2</mn>
+ </msub>
+ </math>
+ </td>
+ <td>
+ <math xmlns="http://www.w3.org/1998/Math/MathML">
+ <msub>
+ <mi>a</mi>
+ <mn>3</mn>
+ </msub>
+ </math>
+ </td>
+ <td>
+ <math xmlns="http://www.w3.org/1998/Math/MathML">
+ <msub>
+ <mi>a</mi>
+ <mn>4</mn>
+ </msub>
+ </math>
+ </td>
+ <td>&nbsp;</td>
+ <td>&nbsp;</td>
+ </tr>
+ <tr>
+ <td>&nbsp;</td>
+ <td>&nbsp;</td>
+ <td>
+ <math xmlns="http://www.w3.org/1998/Math/MathML">
+ <msub>
+ <mi>b</mi>
+ <mn>1</mn>
+ </msub>
+ </math>
+ </td>
+ <td>
+ <math xmlns="http://www.w3.org/1998/Math/MathML">
+ <msub>
+ <mi>b</mi>
+ <mn>2</mn>
+ </msub>
+ </math>
+ </td>
+ <td>
+ <math xmlns="http://www.w3.org/1998/Math/MathML">
+ <msub>
+ <mi>b</mi>
+ <mn>3</mn>
+ </msub>
+ </math>
+ </td>
+ <td>
+ <math xmlns="http://www.w3.org/1998/Math/MathML">
+ <msub>
+ <mi>b</mi>
+ <mn>4</mn>
+ </msub>
+ </math>
+ </td>
+ <td>&nbsp;</td>
+ <td>&nbsp;</td>
+ </tr>
+ <tr>
+ <td>
+ <math xmlns="http://www.w3.org/1998/Math/MathML">
+ <msub>
+ <mi>c</mi>
+ <mn>1</mn>
+ </msub>
+ </math>
+ </td>
+ <td>
+ <math xmlns="http://www.w3.org/1998/Math/MathML">
+ <msub>
+ <mi>d</mi>
+ <mn>1</mn>
+ </msub>
+ </math>
+ </td>
+ <td>0</td>
+ <td>0</td>
+ <td>0</td>
+ <td>0</td>
+ <td>
+ <math xmlns="http://www.w3.org/1998/Math/MathML">
+ <msub>
+ <mi>f</mi>
+ <mn>4</mn>
+ </msub>
+ </math>
+ </td>
+ <td>
+ <math xmlns="http://www.w3.org/1998/Math/MathML">
+ <msub>
+ <mi>e</mi>
+ <mn>4</mn>
+ </msub>
+ </math>
+ </td>
+ </tr>
+ <tr>
+ <td>
+ <math xmlns="http://www.w3.org/1998/Math/MathML">
+ <msub>
+ <mi>c</mi>
+ <mn>2</mn>
+ </msub>
+ </math>
+ </td>
+ <td>
+ <math xmlns="http://www.w3.org/1998/Math/MathML">
+ <msub>
+ <mi>d</mi>
+ <mn>2</mn>
+ </msub>
+ </math>
+ </td>
+ <td>0</td>
+ <td>0</td>
+ <td>0</td>
+ <td>0</td>
+ <td>
+ <math xmlns="http://www.w3.org/1998/Math/MathML">
+ <msub>
+ <mi>f</mi>
+ <mn>3</mn>
+ </msub>
+ </math>
+ </td>
+ <td>
+ <math xmlns="http://www.w3.org/1998/Math/MathML">
+ <msub>
+ <mi>e</mi>
+ <mn>3</mn>
+ </msub>
+ </math>
+ </td>
+ </tr>
+ <tr>
+ <td>
+ <math xmlns="http://www.w3.org/1998/Math/MathML">
+ <msub>
+ <mi>c</mi>
+ <mn>3</mn>
+ </msub>
+ </math>
+ </td>
+ <td>
+ <math xmlns="http://www.w3.org/1998/Math/MathML">
+ <msub>
+ <mi>d</mi>
+ <mn>3</mn>
+ </msub>
+ </math>
+ </td>
+ <td>0</td>
+ <td>0</td>
+ <td>0</td>
+ <td>0</td>
+ <td>
+ <math xmlns="http://www.w3.org/1998/Math/MathML">
+ <msub>
+ <mi>f</mi>
+ <mn>2</mn>
+ </msub>
+ </math>
+ </td>
+ <td>
+ <math xmlns="http://www.w3.org/1998/Math/MathML">
+ <msub>
+ <mi>e</mi>
+ <mn>2</mn>
+ </msub>
+ </math>
+ </td>
+ </tr>
+ <tr>
+ <td>
+ <math xmlns="http://www.w3.org/1998/Math/MathML">
+ <msub>
+ <mi>c</mi>
+ <mn>4</mn>
+ </msub>
+ </math>
+ </td>
+ <td>
+ <math xmlns="http://www.w3.org/1998/Math/MathML">
+ <msub>
+ <mi>d</mi>
+ <mn>4</mn>
+ </msub>
+ </math>
+ </td>
+ <td>0</td>
+ <td>0</td>
+ <td>0</td>
+ <td>0</td>
+ <td>
+ <math xmlns="http://www.w3.org/1998/Math/MathML">
+ <msub>
+ <mi>f</mi>
+ <mn>1</mn>
+ </msub>
+ </math>
+ </td>
+ <td>
+ <math xmlns="http://www.w3.org/1998/Math/MathML">
+ <msub>
+ <mi>e</mi>
+ <mn>1</mn>
+ </msub>
+ </math>
+ </td>
+ </tr>
+ <tr>
+ <td>&nbsp;</td>
+ <td>&nbsp;</td>
+ <td>
+ <math xmlns="http://www.w3.org/1998/Math/MathML">
+ <msub>
+ <mi>h</mi>
+ <mn>4</mn>
+ </msub>
+ </math>
+ </td>
+ <td>
+ <math xmlns="http://www.w3.org/1998/Math/MathML">
+ <msub>
+ <mi>h</mi>
+ <mn>3</mn>
+ </msub>
+ </math>
+ </td>
+ <td>
+ <math xmlns="http://www.w3.org/1998/Math/MathML">
+ <msub>
+ <mi>h</mi>
+ <mn>2</mn>
+ </msub>
+ </math>
+ </td>
+ <td>
+ <math xmlns="http://www.w3.org/1998/Math/MathML">
+ <msub>
+ <mi>h</mi>
+ <mn>1</mn>
+ </msub>
+ </math>
+ </td>
+ <td>&nbsp;</td>
+ <td>&nbsp;</td>
+ </tr>
+ <tr>
+ <td>&nbsp;</td>
+ <td>&nbsp;</td>
+ <td>
+ <math xmlns="http://www.w3.org/1998/Math/MathML">
+ <msub>
+ <mi>g</mi>
+ <mn>4</mn>
+ </msub>
+ </math>
+ </td>
+ <td>
+ <math xmlns="http://www.w3.org/1998/Math/MathML">
+ <msub>
+ <mi>g</mi>
+ <mn>3</mn>
+ </msub>
+ </math>
+ </td>
+ <td>
+ <math xmlns="http://www.w3.org/1998/Math/MathML">
+ <msub>
+ <mi>g</mi>
+ <mn>2</mn>
+ </msub>
+ </math>
+ </td>
+ <td>
+ <math xmlns="http://www.w3.org/1998/Math/MathML">
+ <msub>
+ <mi>g</mi>
+ <mn>1</mn>
+ </msub>
+ </math>
+ </td>
+ <td>&nbsp;</td>
+ <td>&nbsp;</td>
+ </tr>
+ </table>
+ </td>
+ <td>
+ <math xmlns="http://www.w3.org/1998/Math/MathML">
+ <mrow>
+ <mi>Y</mi>
+ <mo>&it;</mo>
+ <mfrac>
+ <msub>
+ <mi>a</mi>
+ <mi>n</mi>
+ </msub>
+ <msub>
+ <mi>b</mi>
+ <mi>n</mi>
+ </msub>
+ </mfrac>
+ <mo>+</mo>
+ <mi>s</mi>
+ <mo>&it;</mo>
+ <mi>W</mi>
+ <mo>&it;</mo>
+ <mfrac>
+ <msub>
+ <mi>c</mi>
+ <mi>n</mi>
+ </msub>
+ <msub>
+ <mi>d</mi>
+ <mi>n</mi>
+ </msub>
+ </mfrac>
+ <mo>=</mo>
+ <mi>Z</mi>
+ <mo>&it;</mo>
+ <mfrac>
+ <msub>
+ <mi>g</mi>
+ <mi>n</mi>
+ </msub>
+ <msub>
+ <mi>h</mi>
+ <mi>n</mi>
+ </msub>
+ </mfrac>
+ <mo>+</mo>
+ <mi>s</mi>
+ <mo>&it;</mo>
+ <mi>X</mi>
+ <mo>&it;</mo>
+ <mfrac>
+ <msub>
+ <mi>e</mi>
+ <mi>n</mi>
+ </msub>
+ <msub>
+ <mi>f</mi>
+ <mi>n</mi>
+ </msub>
+ </mfrac>
+ </mrow>
+ </math>
+ <br />
+ <br />
+ where
+ <br />
+ <br />
+ <math xmlns="http://www.w3.org/1998/Math/MathML">
+ <mrow>
+ <mi>W</mi>
+ <mo>,</mo>
+ <mi>X</mi>
+ <mo>,</mo>
+ <mi>Y</mi>
+ <mo>,</mo>
+ <mi>Z</mi>
+ </mrow>
+ </math>
+ are the common ratios of the arithmetic sequences along the sides of the square of zeros
+ as in the Horseshoe Rule<br />
+ <br />
+ <math xmlns="http://www.w3.org/1998/Math/MathML">
+ <mi>s</mi>
+ </math>
+ is equal to
+ <math xmlns="http://www.w3.org/1998/Math/MathML">
+ <mn>-1</mn>
+ </math>
+ if
+ <math xmlns="http://www.w3.org/1998/Math/MathML">
+ <mi>n</mi>
+ </math>
+ is odd and
+ <math xmlns="http://www.w3.org/1998/Math/MathML">
+ <mn>1</mn>
+ </math>
+ if
+ <math xmlns="http://www.w3.org/1998/Math/MathML">
+ <mi>n</mi>
+ </math>
+ is even
+ </td>
+ <td>
+ As with the Horseshoe Rule, this rule is applicable to squares of zeros of any finite
+ size.<br />
+ <br />
+ The cells that this rule should be targeted at are those along the bottom denoted as
+ <math xmlns="http://www.w3.org/1998/Math/MathML">
+ <msub>
+ <mi>g</mi>
+ <mi>n</mi>
+ </msub>
+ </math>
+ but as with the previous rule, reformulating the math to solve for those cells is left
+ as an exercise to the reader.<br />
+ <br />
+ As with the Long Cross Rule, if any of the cells
+ <math xmlns="http://www.w3.org/1998/Math/MathML">
+ <msub>
+ <mi>a</mi>
+ <mi>n</mi>
+ </msub>
+ </math>
+ go out of bounds of the wall they are considered to be equal to zero. This can occur if
+ there are non-isolated zeros in the input sequence.
+ </td>
+ </tr>
+</table>
+
+ </div>
+</div>
+
+<p>An example of a number wall:</p>
+
+<table class="wall">
+ <tr>
+ <td>&ctdot;</td>
+ <td>1</td>
+ <td>1</td>
+ <td>1</td>
+ <td>1</td>
+ <td>1</td>
+ <td>1</td>
+ <td>1</td>
+ <td>&ctdot;</td>
+ </tr>
+ <tr>
+ <td>&ctdot;</td>
+ <td>64</td>
+ <td>125</td>
+ <td>216</td>
+ <td>343</td>
+ <td>512</td>
+ <td>729</td>
+ <td>1000</td>
+ <td>&ctdot;</td>
+ </tr>
+ <tr>
+ <td>&ctdot;</td>
+ <td>721</td>
+ <td>1801</td>
+ <td>3781</td>
+ <td>7057</td>
+ <td>12097</td>
+ <td>19441</td>
+ <td>29701</td>
+ <td>&ctdot;</td>
+ </tr>
+ <tr>
+ <td>&ctdot;</td>
+ <td>2016</td>
+ <td>4140</td>
+ <td>7344</td>
+ <td>11844</td>
+ <td>17856</td>
+ <td>25596</td>
+ <td>35280</td>
+ <td>&ctdot;</td>
+ </tr>
+ <tr>
+ <td>&ctdot;</td>
+ <td>1296</td>
+ <td>1296</td>
+ <td>1296</td>
+ <td>1296</td>
+ <td>1296</td>
+ <td>1296</td>
+ <td>1296</td>
+ <td>&ctdot;</td>
+ </tr>
+ <tr>
+ <td>&ctdot;</td>
+ <td>0</td>
+ <td>0</td>
+ <td>0</td>
+ <td>0</td>
+ <td>0</td>
+ <td>0</td>
+ <td>0</td>
+ <td>&ctdot;</td>
+ </tr>
+ <tr>
+ <td>&utdot;</td>
+ <td>&vellip;</td>
+ <td>&vellip;</td>
+ <td>&vellip;</td>
+ <td>&vellip;</td>
+ <td>&vellip;</td>
+ <td>&vellip;</td>
+ <td>&vellip;</td>
+ <td>&dtdot;</td>
+ </tr>
+</table>
+
+<p>The above excerpt was taken from the number wall generated by the sequence of integer cubes. In
+this case, the wall essentially ends in a row of all zeros as all subsequent rows below that will
+also be zeros. Note however that this does not always happen.</p>
+
+<p>A property of number walls is that zeros will always be arranged in squares. A singular isolated
+zero is considered to be a 1x1 square and a wall ending in rows of zeros as above is considered to
+be the start of a square of zeros of infinite size. This property can be used to supplement the
+calculation rules.</p>
+
+{% endblock -%}
+
+
diff --git a/project/templates/steelman.xhtml b/project/templates/steelman.xhtml
index 94a7721..77ed0b6 100644
--- a/project/templates/steelman.xhtml
+++ b/project/templates/steelman.xhtml
@@ -13,6 +13,7 @@
{%- block style %}
<link href="/css/steelman.css" rel="stylesheet" />
+ <link href="/css/accordian.css" rel="stylesheet" />
{% endblock -%}
generated by cgit (git 2.30.0) at 2024-12-22 12:47:22 +0000