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authorMarshall Lochbaum <mwlochbaum@gmail.com>2022-05-31 21:56:13 -0400
committerMarshall Lochbaum <mwlochbaum@gmail.com>2022-05-31 21:56:13 -0400
commit6078db236e6c34788371576bb51410cf8298b583 (patch)
tree0e62588f942289a4fef397b19c5f06c5591f9d1a /docs/doc/array.html
parentf0d58667085fee808a59b9c9e7667ad18d074cdf (diff)
Define cells in the documentation
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<p>The total number of elements in an array is its <strong>bound</strong>, and can be found using <a href="reshape.html">Deshape</a> with <code><span class='Function'>≠</span><span class='Modifier2'>∘</span><span class='Function'>⥊</span></code>, or by multiplying all the lengths in the shape. An array of rank 0, which always contains exactly one element, is called a <a href="enclose.html#whats-a-unit"><strong>unit</strong></a>, while an array of rank 1 is called a <strong>list</strong> and an array of rank 2 is called a <strong>table</strong>.</p>
<h2 id="elements"><a class="header" href="#elements">Elements</a></h2>
<p>Any BQN value can be used as an array element, including another array (BQN, as a dynamically-typed language, doesn't restrict the types that can be used in one context without a good reason). However, BQN arrays are restricted relative to other array models. Frameworks like NumPy or Julia have mutable arrays, so that the value of an element can be changed after the array is created. This allows an array to be its own element, by creating an array and then inserting it into itself. This would be unnatural in BQN, where an array can only be formed from elements that already exist. In BQN only operations and namespaces are <a href="lexical.html#mutation">mutable</a>.</p>
+<h2 id="cells"><a class="header" href="#cells">Cells</a></h2>
+<p>The contents of an array are its elements, but it also makes sense to split up an array into subarrays of elements called cells. The most important kind of cell, a <strong>major cell</strong> consists of all the elements that have indices beginning with some particular index <code><span class='Value'>i</span></code>. For this to make sense, <code><span class='Value'>i</span></code> must be between <code><span class='Number'>0</span></code> and the length <code><span class='Value'>l</span></code> of the array's first axis, so that there are <code><span class='Value'>l</span></code> major cells each identified by an index.</p>
+<a class="replLink" title="Open in the REPL" target="_blank" href="https://mlochbaum.github.io/BQN/try.html#code=MuKAvzPigL80IMOX4oycIDHigL814oC/OOKAvzExCgoxIOKKjyAy4oC/M+KAvzQgw5fijJwgMeKAvzXigL844oC/MTEgICMgTWFqb3IgY2VsbCAx">↗️</a><pre> <span class='Number'>2</span><span class='Ligature'>‿</span><span class='Number'>3</span><span class='Ligature'>‿</span><span class='Number'>4</span> <span class='Function'>×</span><span class='Modifier'>⌜</span> <span class='Number'>1</span><span class='Ligature'>‿</span><span class='Number'>5</span><span class='Ligature'>‿</span><span class='Number'>8</span><span class='Ligature'>‿</span><span class='Number'>11</span>
+┌─
+╵ 2 10 16 22
+ 3 15 24 33
+ 4 20 32 44
+ ┘
+
+ <span class='Number'>1</span> <span class='Function'>⊏</span> <span class='Number'>2</span><span class='Ligature'>‿</span><span class='Number'>3</span><span class='Ligature'>‿</span><span class='Number'>4</span> <span class='Function'>×</span><span class='Modifier'>⌜</span> <span class='Number'>1</span><span class='Ligature'>‿</span><span class='Number'>5</span><span class='Ligature'>‿</span><span class='Number'>8</span><span class='Ligature'>‿</span><span class='Number'>11</span> <span class='Comment'># Major cell 1
+</span>⟨ 3 15 24 33 ⟩
+</pre>
+<p>A major cell still has an array structure: it retains all the axes of the original array other than the first. So it has its own major cells, identified by the index <code><span class='Value'>i</span></code> of the original major cell and <code><span class='Value'>j</span></code> within it. These are also cells of the original array. Generalizing, a <strong>cell</strong> with index list <code><span class='Value'>l</span></code> is defined to be the array of all elements whose indices begin with <code><span class='Value'>l</span></code>. In an array with rank <code><span class='Value'>n</span></code>, the cell rank is <code><span class='Value'>n</span><span class='Function'>-≠</span><span class='Value'>l</span></code>, and cells grouped using this rank. An <code><span class='Value'>n</span></code>-cell mst have an empty cell index, so that it includes all elements—it's the entire array! An <code><span class='Value'>n</span><span class='Function'>-</span><span class='Number'>1</span></code> cell, also called a ¯1-cell, is a major cell. A 0-cell has an index of length <code><span class='Value'>n</span></code>, and contains a single element.</p>
+<p>Cells are the center of the <a href="leading.html">leading axis model</a> used to structure many array primitives.</p>
<h2 id="properties"><a class="header" href="#properties">Properties</a></h2>
<p>Summarizing, the values needed to define an array are its rank (the number of axes), its shape (the number of positions along each axis), and the value of each element (that is, at each combination of positions). Two arrays <a href="match.html">match</a> when all these values match.</p>
<p>If the rank is considered to be part of the shape, as it is when the shape is a BQN list, then the array is defined by its shape and element list—from <a href="reshape.html">deshape</a>.</p>