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<div class="section" id="Accessing-Data-Along-Multiple-Dimensions-in-an-Array">
<h1>Accessing Data Along Multiple Dimensions in an Array<a class="headerlink" href="#Accessing-Data-Along-Multiple-Dimensions-in-an-Array" title="Permalink to this headline"></a></h1>
<p>In this section, we will:</p>
<ul class="simple">
<li><p>Define the “dimensionality” of an array.</p></li>
<li><p>Discuss the usefulness of ND-arrays.</p></li>
<li><p>Introduce the indexing and slicing scheme for accessing a multi-dimensional array’s contents</p></li>
</ul>
<p>We will encounter arrays of varying dimensionalities:</p>
<div class="highlight-python notranslate"><div class="highlight"><pre><span></span><span class="c1"># A 0-D array</span>
<span class="n">np</span><span class="o">.</span><span class="n">array</span><span class="p">(</span><span class="mi">8</span><span class="p">)</span>
<span class="c1"># A 1-D array, shape-(3,)</span>
<span class="n">np</span><span class="o">.</span><span class="n">array</span><span class="p">([</span><span class="mf">2.3</span><span class="p">,</span> <span class="mf">0.1</span><span class="p">,</span> <span class="o">-</span><span class="mf">9.1</span><span class="p">])</span>
<span class="c1"># A 2-D array, shape-(3, 2)</span>
<span class="n">np</span><span class="o">.</span><span class="n">array</span><span class="p">([[</span><span class="mi">93</span><span class="p">,</span> <span class="mi">95</span><span class="p">],</span>
<span class="p">[</span><span class="mi">84</span><span class="p">,</span> <span class="mi">100</span><span class="p">],</span>
<span class="p">[</span><span class="mi">99</span><span class="p">,</span> <span class="mi">87</span><span class="p">]])</span>
<span class="c1"># A 3-D array, shape-(2, 2, 2)</span>
<span class="n">np</span><span class="o">.</span><span class="n">array</span><span class="p">([[[</span><span class="mi">0</span><span class="p">,</span> <span class="mi">1</span><span class="p">],</span>
<span class="p">[</span><span class="mi">2</span><span class="p">,</span> <span class="mi">3</span><span class="p">]],</span>
<span class="p">[[</span><span class="mi">4</span><span class="p">,</span> <span class="mi">5</span><span class="p">],</span>
<span class="p">[</span><span class="mi">6</span><span class="p">,</span> <span class="mi">7</span><span class="p">]]])</span>
</pre></div>
</div>
<p>Similar to Python’s sequences, we use 0-based indices and slicing to access the content of an array. However, we must specify an index/slice for <em>each</em> dimension of an array:</p>
<div class="highlight-python notranslate"><div class="highlight"><pre><span></span><span class="gp">>>> </span><span class="kn">import</span> <span class="nn">numpy</span> <span class="k">as</span> <span class="nn">np</span>
<span class="go"># A 3-D array</span>
<span class="gp">>>> </span><span class="n">x</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">array</span><span class="p">([[[</span><span class="mi">0</span><span class="p">,</span> <span class="mi">1</span><span class="p">],</span>
<span class="gp">... </span> <span class="p">[</span><span class="mi">2</span><span class="p">,</span> <span class="mi">3</span><span class="p">]],</span>
<span class="gp">...</span>
<span class="gp">... </span> <span class="p">[[</span><span class="mi">4</span><span class="p">,</span> <span class="mi">5</span><span class="p">],</span>
<span class="gp">... </span> <span class="p">[</span><span class="mi">6</span><span class="p">,</span> <span class="mi">7</span><span class="p">]]])</span>
<span class="go"># get: sheet-0, both rows, flip order of columns</span>
<span class="gp">>>> </span><span class="n">x</span><span class="p">[</span><span class="mi">0</span><span class="p">,</span> <span class="p">:,</span> <span class="p">::</span><span class="o">-</span><span class="mi">1</span><span class="p">]</span>
<span class="go">array([[1, 0],</span>
<span class="go"> [3, 2]])</span>
</pre></div>
</div>
<div class="section" id="One-dimensional-Arrays">
<h2>One-dimensional Arrays<a class="headerlink" href="#One-dimensional-Arrays" title="Permalink to this headline"></a></h2>
<p>Let’s begin our discussion by constructing a simple ND-array containing three floating-point numbers.</p>
<div class="highlight-python notranslate"><div class="highlight"><pre><span></span><span class="gp">>>> </span><span class="n">simple_array</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">array</span><span class="p">([</span><span class="mf">2.3</span><span class="p">,</span> <span class="mf">0.1</span><span class="p">,</span> <span class="o">-</span><span class="mf">9.1</span><span class="p">])</span>
</pre></div>
</div>
<p>This array supports the same indexing scheme as Python’s sequences (lists, tuples, and strings):</p>
<div class="highlight-none notranslate"><div class="highlight"><pre><span></span>+------+------+------+
| 2.3 | 0.1 | -9.1 |
+------+------+------+
0 1 2
-3 -2 -1
</pre></div>
</div>
<p>The first row of numbers gives the position of the indices 0…3 in the array; the second row gives the corresponding negative indices. The slice from <span class="math notranslate nohighlight">\(i\)</span> to <span class="math notranslate nohighlight">\(j\)</span> returns an array containing of all numbers between the edges labeled <span class="math notranslate nohighlight">\(i\)</span> and <span class="math notranslate nohighlight">\(j\)</span>, respectively:</p>
<div class="highlight-python notranslate"><div class="highlight"><pre><span></span><span class="gp">>>> </span><span class="n">simple_array</span><span class="p">[</span><span class="mi">0</span><span class="p">]</span>
<span class="go">2.3</span>
<span class="gp">>>> </span><span class="n">simple_array</span><span class="p">[</span><span class="o">-</span><span class="mi">2</span><span class="p">]</span>
<span class="go">0.1</span>
<span class="gp">>>> </span><span class="n">simple_array</span><span class="p">[</span><span class="mi">1</span><span class="p">:</span><span class="mi">3</span><span class="p">]</span>
<span class="go">array([ 0.1, -9.1])</span>
<span class="gp">>>> </span><span class="n">simple_array</span><span class="p">[</span><span class="mi">3</span><span class="p">]</span>
<span class="go">IndexError: index 3 is out of bounds for axis 0 with size 3</span>
</pre></div>
</div>
<p>Given this indexing scheme, only <em>one</em> integer is needed to specify a unique entry in the array. Similarly only <em>one</em> slice is needed to uniquely specify a subsequence of entries in the array. For this reason, we say that this is a <em>1-dimensional array</em>. In general, the <em>dimensionality</em> of an array specifies the number of indices that are required to uniquely specify one of its entries.</p>
<div class="admonition note">
<p class="admonition-title fa fa-exclamation-circle"><strong>Definition</strong>:</p>
<p>The <strong>dimensionality</strong> of an array specifies the number of indices that are required to uniquely specify one of its entries.</p>
</div>
<p>This definition of dimensionality is common far beyond NumPy; one must use three numbers to uniquely specify a point in physical space, which is why it is said that space consists of three dimensions.</p>
</div>
<div class="section" id="Two-dimensional-Arrays">
<h2>Two-dimensional Arrays<a class="headerlink" href="#Two-dimensional-Arrays" title="Permalink to this headline"></a></h2>
<p>Before proceeding further down the path of high-dimensional arrays, let’s briefly consider a very simple dataset where the desire to access the data along multiple dimensions is manifestly desirable. Consider the following table from a gradebook:</p>
<table class="docutils align-default">
<colgroup>
<col style="width: 24%" />
<col style="width: 36%" />
<col style="width: 39%" />
</colgroup>
<thead>
<tr class="row-odd"><th class="head"></th>
<th class="head"><p>Exam 1 (%)</p></th>
<th class="head"><p>Exam 2 (%)</p></th>
</tr>
</thead>
<tbody>
<tr class="row-even"><td><p>Ashley</p></td>
<td><p><span class="math notranslate nohighlight">\(93\)</span></p></td>
<td><p><span class="math notranslate nohighlight">\(95\)</span></p></td>
</tr>
<tr class="row-odd"><td><p>Brad</p></td>
<td><p><span class="math notranslate nohighlight">\(84\)</span></p></td>
<td><p><span class="math notranslate nohighlight">\(100\)</span></p></td>
</tr>
<tr class="row-even"><td><p>Cassie</p></td>
<td><p><span class="math notranslate nohighlight">\(99\)</span></p></td>
<td><p><span class="math notranslate nohighlight">\(87\)</span></p></td>
</tr>
</tbody>
</table>
<p>This dataset contains 6 grade-values. It is almost immediately clear that storing these in a 1-dimensional array is not ideal:</p>
<div class="highlight-python notranslate"><div class="highlight"><pre><span></span><span class="c1"># using a 1-dimensional array to store the grades</span>
<span class="o">>>></span> <span class="n">grades</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">array</span><span class="p">([</span><span class="mi">93</span><span class="p">,</span> <span class="mi">95</span><span class="p">,</span> <span class="mi">84</span><span class="p">,</span> <span class="mi">100</span><span class="p">,</span> <span class="mi">99</span><span class="p">,</span> <span class="mi">87</span><span class="p">])</span>
</pre></div>
</div>
<p>While no data has been lost, accessing this data using a single index is less than convenient; we want to be able to specify both the student and the exam when accessing a grade - it is natural to ascribe <em>two dimensions</em> to this data. Let’s construct a 2D array containing these grades:</p>
<div class="highlight-python notranslate"><div class="highlight"><pre><span></span><span class="c1"># using a 2-dimensional array to store the grades</span>
<span class="o">>>></span> <span class="n">grades</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">array</span><span class="p">([[</span><span class="mi">93</span><span class="p">,</span> <span class="mi">95</span><span class="p">],</span>
<span class="o">...</span> <span class="p">[</span><span class="mi">84</span><span class="p">,</span> <span class="mi">100</span><span class="p">],</span>
<span class="o">...</span> <span class="p">[</span><span class="mi">99</span><span class="p">,</span> <span class="mi">87</span><span class="p">]])</span>
</pre></div>
</div>
<p>NumPy is able to see the repeated structure among the list-of-lists-of-numbers passed to <code class="docutils literal notranslate"><span class="pre">np.array</span></code>, and resolve the two dimensions of data, which we deem the ‘student’ dimension and the ‘exam’ dimension, respectively.</p>
<div class="admonition warning">
<p class="admonition-title fa fa-exclamation-circle"><strong>Axis vs Dimension</strong>:</p>
<p>Although NumPy does formally recognize the concept of dimensionality precisely in the way that it is discussed here, its documentation refers to an individual dimension of an array as an <strong>axis</strong>. Thus you will see “axes” (pronounced “aks-ēz”) used in place of “dimensions”; however, they mean the same thing.</p>
</div>
<p>NumPy specifies the row-axis (students) of a 2D array as “axis-0” and the column-axis (exams) as axis-1. You must now provide <em>two</em> indices, one for each axis (dimension), to uniquely specify an element in this 2D array; the first number specifies an index along axis-0, the second specifies an index along axis-1. The zero-based indexing schema that we reviewed earlier applies to each axis of the ND-array:</p>
<div class="highlight-none notranslate"><div class="highlight"><pre><span></span> -- axis-1 ->
-2 -1
0 1
| +---+---+
| -3, 0 |93 | 95|
| +---+---+
axis-0 -2, 1 |84 |100|
| +---+---+
| -1, 2 |99 | 87|
V +---+---+
</pre></div>
</div>
<p>Because <code class="docutils literal notranslate"><span class="pre">grades</span></code> has three entries along axis-0 and two entries along axis-1, it has a “shape” of <code class="docutils literal notranslate"><span class="pre">(3,</span> <span class="pre">2)</span></code>.</p>
<div class="highlight-python notranslate"><div class="highlight"><pre><span></span><span class="gp">>>> </span><span class="n">grades</span><span class="o">.</span><span class="n">shape</span>
<span class="go">(3, 2)</span>
</pre></div>
</div>
<div class="section" id="Integer-Indexing">
<h3>Integer Indexing<a class="headerlink" href="#Integer-Indexing" title="Permalink to this headline"></a></h3>
<p>Thus, if we want to access Brad’s (item-1 along axis-0) score for Exam 1 (item-0 along axis-1) we simply specify:</p>
<div class="highlight-python notranslate"><div class="highlight"><pre><span></span><span class="c1"># providing two numbers to access an element</span>
<span class="c1"># in a 2D-array</span>
<span class="o">>>></span> <span class="n">grades</span><span class="p">[</span><span class="mi">1</span><span class="p">,</span> <span class="mi">0</span><span class="p">]</span> <span class="c1"># Brad's score on Exam 1</span>
<span class="mi">84</span>
<span class="c1"># negative indices work as with lists/tuples/strings</span>
<span class="o">>>></span> <span class="n">grades</span><span class="p">[</span><span class="o">-</span><span class="mi">2</span><span class="p">,</span> <span class="mi">0</span><span class="p">]</span> <span class="c1"># Brad's score on Exam 1</span>
<span class="mi">84</span>
</pre></div>
</div>
</div>
<div class="section" id="Slice-Indexing">
<h3>Slice Indexing<a class="headerlink" href="#Slice-Indexing" title="Permalink to this headline"></a></h3>
<p>We can also uses <em>slices</em> to access subsequences of our data. Suppose we want the scores of all the students for Exam 2. We can slice from 0 through 3 along axis-0 (refer to the indexing diagram in the previous section) to include all the students, and specify index 1 on axis-1 to select Exam 2:</p>
<div class="highlight-python notranslate"><div class="highlight"><pre><span></span><span class="gp">>>> </span><span class="n">grades</span><span class="p">[</span><span class="mi">0</span><span class="p">:</span><span class="mi">3</span><span class="p">,</span> <span class="mi">1</span><span class="p">]</span> <span class="c1"># Exam 2 scores for all students</span>
<span class="go">array([ 95, 100, 87])</span>
</pre></div>
</div>
<p>As with Python sequences, you can specify an “empty” slice to include all possible entries along an axis, by default: <code class="docutils literal notranslate"><span class="pre">grades[:,</span> <span class="pre">1]</span></code> is equivalent to <code class="docutils literal notranslate"><span class="pre">grades[0:3,</span> <span class="pre">1]</span></code>, in this instance. More generally, withholding either the ‘start’ or ‘stop’ value in a slice will result in the use smallest or largest valid index, respectively:</p>
<div class="highlight-python notranslate"><div class="highlight"><pre><span></span><span class="gp">>>> </span><span class="n">grades</span><span class="p">[</span><span class="mi">1</span><span class="p">:,</span> <span class="mi">1</span><span class="p">]</span> <span class="c1"># equivalent to `grades[1:3, 1]</span>
<span class="go">array([ 100, 87])</span>
<span class="gp">>>> </span><span class="n">grades</span><span class="p">[:,</span> <span class="p">:</span><span class="mi">1</span><span class="p">]</span> <span class="c1"># equivalent to `grades[0:3, 0:1]</span>
<span class="go">array([[93],</span>
<span class="go"> [84],</span>
<span class="go"> [99]])</span>
</pre></div>
</div>
<p>The output of <code class="docutils literal notranslate"><span class="pre">grades[:,</span> <span class="pre">:1]</span></code> might look somewhat funny. Because the axis-1 slice only includes one column of numbers, the shape of the resulting array is (3, 1). 0 is thus only valid (non-negative) index for axis-1, since there is only one column to specify in the array.</p>
<p>You can also supply a “step” value to the slice. <code class="docutils literal notranslate"><span class="pre">grades[::-1,</span> <span class="pre">:]</span></code> will returns the array of grades with the student-axis flipped (reverse-alphabetical order).</p>
</div>
<div class="section" id="Negative-Indices">
<h3>Negative Indices<a class="headerlink" href="#Negative-Indices" title="Permalink to this headline"></a></h3>
<p>As indicated above, negative indices are valid too and are quite useful. If we want to access the scores of the latest exam for all of the students, you can specify:</p>
<div class="highlight-python notranslate"><div class="highlight"><pre><span></span><span class="c1"># using a negative index and a slice</span>
<span class="o">>>></span> <span class="n">grades</span><span class="p">[:,</span> <span class="o">-</span><span class="mi">1</span><span class="p">]</span> <span class="c1"># Latest exam scores (Exam 2), for all students</span>
<span class="n">array</span><span class="p">([</span> <span class="mi">95</span><span class="p">,</span> <span class="mi">100</span><span class="p">,</span> <span class="mi">87</span><span class="p">])</span>
</pre></div>
</div>
<p>Note the value of using the negative index is that it will always provide you with the latest exam score - you need not check how many exams the students have taken.</p>
</div>
<div class="section" id="Supplying-Fewer-Indices-Than-Dimensions">
<h3>Supplying Fewer Indices Than Dimensions<a class="headerlink" href="#Supplying-Fewer-Indices-Than-Dimensions" title="Permalink to this headline"></a></h3>
<p>What happens if we only supply one index to our array? It may be surprising that <code class="docutils literal notranslate"><span class="pre">grades[0]</span></code> does not throw an error since we are specifying only one index to access data from a 2-dimensional array. Instead, NumPy it will return all of the exam scores for student-0 (Ashley):</p>
<div class="highlight-python notranslate"><div class="highlight"><pre><span></span><span class="gp">>>> </span><span class="n">grades</span><span class="p">[</span><span class="mi">0</span><span class="p">]</span>
<span class="go">array([ 93, 95])</span>
</pre></div>
</div>
<p>This is because NumPy will automatically insert trailing slices for you if you don’t provide as many indices as there are dimensions for your array. <code class="docutils literal notranslate"><span class="pre">grades[0]</span></code> was treated as <code class="docutils literal notranslate"><span class="pre">grades[0,</span> <span class="pre">:]</span></code>.</p>
<div class="admonition note">
<p>Suppose you have an <span class="math notranslate nohighlight">\(N\)</span>-dimensional array, and only provide <span class="math notranslate nohighlight">\(j\)</span> indices for the array; NumPy will automatically insert <span class="math notranslate nohighlight">\(N-j\)</span> trailing slices for you. In the case that <span class="math notranslate nohighlight">\(N=5\)</span> and <span class="math notranslate nohighlight">\(j=3\)</span>, <code class="docutils literal notranslate"><span class="pre">d5_array[0,</span> <span class="pre">0,</span> <span class="pre">0]</span></code> is treated as <code class="docutils literal notranslate"><span class="pre">d5_array[0,</span> <span class="pre">0,</span> <span class="pre">0,</span> <span class="pre">:,</span> <span class="pre">:]</span></code></p>
</div>
<p>Thus far, we have discussed some rules for accessing data in arrays, all of which fall into the category that is designated <a class="reference external" href="https://numpy.org/doc/stable/reference/arrays.indexing.html#basic-slicing-and-indexing">“basic indexing”</a> by the NumPy documentation. We will discuss the details of basic indexing and of <a class="reference external" href="https://numpy.org/doc/stable/reference/arrays.indexing.html#advanced-indexing">“advanced indexing”</a>, in full, in a later section. Note, however, that all of the indexing/slicing
reviewed here produces a “view” of the original array. That is, <em>no data is copied</em> when you index into an array using integer indices and/or slices. Recall that slicing lists and tuples <em>do</em> produce copies of the data.</p>
<div class="admonition warning">
<p class="admonition-title fa fa-exclamation-circle"><strong>FYI</strong>:</p>
<p>Keeping track of the meaning of an array’s various dimensions can quickly become unwieldy when working with real datasets. <a class="reference external" href="http://xarray.pydata.org/en/stable/">xarray</a> is a Python library that provides functionality comparable to NumPy, but allows users provide <em>explicit labels</em> for an array’s dimensions; that is, you can <em>name</em> each dimension. Using an <code class="docutils literal notranslate"><span class="pre">xarray</span></code> to select Brad’s scores could look like <code class="docutils literal notranslate"><span class="pre">grades.sel(student='Brad')</span></code>, for instance. This is a valuable library to look into at
your leisure.</p>
</div>
</div>
</div>
<div class="section" id="N-dimensional-Arrays">
<h2>N-dimensional Arrays<a class="headerlink" href="#N-dimensional-Arrays" title="Permalink to this headline"></a></h2>
<p>Let’s build up some intuition for arrays with a dimensionality higher than 2. The following code creates a 3-dimensional array:</p>
<div class="highlight-python notranslate"><div class="highlight"><pre><span></span><span class="c1"># a 3D array, shape-(2, 2, 2)</span>
<span class="o">>>></span> <span class="n">d3_array</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">array</span><span class="p">([[[</span><span class="mi">0</span><span class="p">,</span> <span class="mi">1</span><span class="p">],</span>
<span class="o">...</span> <span class="p">[</span><span class="mi">2</span><span class="p">,</span> <span class="mi">3</span><span class="p">]],</span>
<span class="o">...</span>
<span class="o">...</span> <span class="p">[[</span><span class="mi">4</span><span class="p">,</span> <span class="mi">5</span><span class="p">],</span>
<span class="o">...</span> <span class="p">[</span><span class="mi">6</span><span class="p">,</span> <span class="mi">7</span><span class="p">]]])</span>
</pre></div>
</div>
<p>You can think of axis-0 denoting which of the 2x2 “sheets” to select from. Then axis-1 specifies the row along the sheets, and axis-2 the column within the row:</p>
<p><strong>Depicting the layout of a 3D array</strong></p>
<div class="highlight-none notranslate"><div class="highlight"><pre><span></span>sheet 0:
[0, 1]
[2, 3]
sheet 1:
[4, 5]
[6, 7]
</pre></div>
</div>
<div class="highlight-none notranslate"><div class="highlight"><pre><span></span> | -- axis-2 ->
| |
| axis-1 [0, 1]
| | [2, 3]
| V
axis-0
| -- axis-2 ->
| |
| axis-1 [4, 5]
| | [6, 7]
V V
</pre></div>
</div>
<p>Thus <code class="docutils literal notranslate"><span class="pre">d3_array[0,</span> <span class="pre">1,</span> <span class="pre">0]</span></code> specifies the element residing in sheet-0, at row-1 and column-0:</p>
<div class="highlight-python notranslate"><div class="highlight"><pre><span></span><span class="c1"># retrieving a single element from a 3D-array</span>
<span class="o">>>></span> <span class="n">d3_array</span><span class="p">[</span><span class="mi">0</span><span class="p">,</span> <span class="mi">1</span><span class="p">,</span> <span class="mi">0</span><span class="p">]</span>
<span class="mi">2</span>
</pre></div>
</div>
<p><code class="docutils literal notranslate"><span class="pre">d3_array[:,</span> <span class="pre">0,</span> <span class="pre">0]</span></code> specifies the elements in row-0 and column-0 of <strong>both</strong> sheets:</p>
<div class="highlight-python notranslate"><div class="highlight"><pre><span></span><span class="c1"># retrieving a 1D sub-array from a 3D-array</span>
<span class="o">>>></span> <span class="n">d3_array</span><span class="p">[:,</span> <span class="mi">0</span><span class="p">,</span> <span class="mi">0</span><span class="p">]</span>
<span class="n">array</span><span class="p">([</span><span class="mi">0</span><span class="p">,</span> <span class="mi">4</span><span class="p">])</span>
</pre></div>
</div>
<p><code class="docutils literal notranslate"><span class="pre">d3_array[1]</span></code>, which recall is shorthand for <code class="docutils literal notranslate"><span class="pre">d3_array[1,</span> <span class="pre">:,</span> <span class="pre">:]</span></code>, selects both rows and both columns of sheet-1:</p>
<div class="highlight-python notranslate"><div class="highlight"><pre><span></span><span class="c1"># retrieving a 2D sub-array from a 3D-array</span>
<span class="o">>>></span> <span class="n">d3_array</span><span class="p">[</span><span class="mi">1</span><span class="p">]</span>
<span class="n">array</span><span class="p">([[</span><span class="mi">4</span><span class="p">,</span> <span class="mi">5</span><span class="p">],</span>
<span class="p">[</span><span class="mi">6</span><span class="p">,</span> <span class="mi">7</span><span class="p">]])</span>
</pre></div>
</div>
<p>In four dimensions, one can think of “<em>stacks</em> of sheets with rows and columns” where axis-0 selects the stack of sheets you are working with, axis-1 chooses the sheet, axis-2 chooses the row, and axis-3 chooses the column. Extrapolating to higher dimensions (“collections of stacks of sheets …”) continues in the same tedious fashion.</p>
<div class="admonition note">
<p class="admonition-title fa fa-exclamation-circle"><strong>Reading Comprehension: Multi-dimensional Indexing</strong></p>
<p>Given the 3D, shape-(3, 3, 3) array:</p>
<div class="highlight-python notranslate"><div class="highlight"><pre><span></span><span class="gp">>>> </span><span class="n">arr</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">array</span><span class="p">([[[</span> <span class="mi">0</span><span class="p">,</span> <span class="mi">1</span><span class="p">,</span> <span class="mi">2</span><span class="p">],</span>
<span class="gp">... </span> <span class="p">[</span> <span class="mi">3</span><span class="p">,</span> <span class="mi">4</span><span class="p">,</span> <span class="mi">5</span><span class="p">],</span>
<span class="gp">... </span> <span class="p">[</span> <span class="mi">6</span><span class="p">,</span> <span class="mi">7</span><span class="p">,</span> <span class="mi">8</span><span class="p">]],</span>
<span class="gp">...</span>
<span class="gp">... </span> <span class="p">[[</span> <span class="mi">9</span><span class="p">,</span> <span class="mi">10</span><span class="p">,</span> <span class="mi">11</span><span class="p">],</span>
<span class="gp">... </span> <span class="p">[</span><span class="mi">12</span><span class="p">,</span> <span class="mi">13</span><span class="p">,</span> <span class="mi">14</span><span class="p">],</span>
<span class="gp">... </span> <span class="p">[</span><span class="mi">15</span><span class="p">,</span> <span class="mi">16</span><span class="p">,</span> <span class="mi">17</span><span class="p">]],</span>
<span class="gp">...</span>
<span class="gp">... </span> <span class="p">[[</span><span class="mi">18</span><span class="p">,</span> <span class="mi">19</span><span class="p">,</span> <span class="mi">20</span><span class="p">],</span>
<span class="gp">... </span> <span class="p">[</span><span class="mi">21</span><span class="p">,</span> <span class="mi">22</span><span class="p">,</span> <span class="mi">23</span><span class="p">],</span>
<span class="gp">... </span> <span class="p">[</span><span class="mi">24</span><span class="p">,</span> <span class="mi">25</span><span class="p">,</span> <span class="mi">26</span><span class="p">]]])</span>
</pre></div>
</div>
<p>Index into the array to produce the following results</p>
<div class="highlight-python notranslate"><div class="highlight"><pre><span></span><span class="c1">#1</span>
<span class="n">array</span><span class="p">([[</span> <span class="mi">2</span><span class="p">,</span> <span class="mi">5</span><span class="p">,</span> <span class="mi">8</span><span class="p">],</span>
<span class="p">[</span><span class="mi">11</span><span class="p">,</span> <span class="mi">14</span><span class="p">,</span> <span class="mi">17</span><span class="p">],</span>
<span class="p">[</span><span class="mi">20</span><span class="p">,</span> <span class="mi">23</span><span class="p">,</span> <span class="mi">26</span><span class="p">]])</span>
<span class="c1">#2</span>
<span class="n">array</span><span class="p">([[</span> <span class="mi">3</span><span class="p">,</span> <span class="mi">4</span><span class="p">,</span> <span class="mi">5</span><span class="p">],</span>
<span class="p">[</span><span class="mi">12</span><span class="p">,</span> <span class="mi">13</span><span class="p">,</span> <span class="mi">14</span><span class="p">]])</span>
<span class="c1">#3</span>
<span class="n">array</span><span class="p">([</span><span class="mi">2</span><span class="p">,</span> <span class="mi">5</span><span class="p">])</span>
<span class="c1">#4</span>
<span class="n">array</span><span class="p">([[</span><span class="mi">11</span><span class="p">,</span> <span class="mi">10</span><span class="p">,</span> <span class="mi">9</span><span class="p">],</span>
<span class="p">[</span><span class="mi">14</span><span class="p">,</span> <span class="mi">13</span><span class="p">,</span> <span class="mi">12</span><span class="p">],</span>
<span class="p">[</span><span class="mi">17</span><span class="p">,</span> <span class="mi">16</span><span class="p">,</span> <span class="mi">15</span><span class="p">]])</span>
</pre></div>
</div>
</div>
</div>
<div class="section" id="Zero-dimensional-Arrays">
<h2>Zero-dimensional Arrays<a class="headerlink" href="#Zero-dimensional-Arrays" title="Permalink to this headline"></a></h2>
<p>A zero dimensional array is simply a single number (a.k.a. a scalar value):</p>
<div class="highlight-python notranslate"><div class="highlight"><pre><span></span><span class="c1"># creating a 0-dimensional array</span>
<span class="o">>>></span> <span class="n">x</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">array</span><span class="p">(</span><span class="mf">15.2</span><span class="p">)</span>
</pre></div>
</div>
<p>This is <em>not</em> equivalent to a length-1 1D-array: <code class="docutils literal notranslate"><span class="pre">np.array([15.2])</span></code>. According to our definition of dimensionality, <em>zero</em> numbers are required to index into a 0-D array as it is unnecessary to provide an identifier for a standalone number. Thus you cannot index into a 0-D array.</p>
<div class="highlight-python notranslate"><div class="highlight"><pre><span></span><span class="c1"># you cannot index into a 0-D array</span>
<span class="o">>>></span> <span class="n">x</span><span class="p">[</span><span class="mi">0</span><span class="p">]</span>
<span class="o">---------------------------------------------------------------------------</span>
<span class="ne">IndexError</span> <span class="n">Traceback</span> <span class="p">(</span><span class="n">most</span> <span class="n">recent</span> <span class="n">call</span> <span class="n">last</span><span class="p">)</span>
<span class="o"><</span><span class="n">ipython</span><span class="o">-</span><span class="nb">input</span><span class="o">-</span><span class="mi">10</span><span class="o">-</span><span class="mi">2</span><span class="n">f755f117ac9</span><span class="o">></span> <span class="ow">in</span> <span class="o"><</span><span class="n">module</span><span class="o">></span><span class="p">()</span>
<span class="o">----></span> <span class="mi">1</span> <span class="n">x</span><span class="p">[</span><span class="mi">0</span><span class="p">]</span>
<span class="ne">IndexError</span><span class="p">:</span> <span class="n">too</span> <span class="n">many</span> <span class="n">indices</span> <span class="k">for</span> <span class="n">array</span>
</pre></div>
</div>
<p>You must use the syntax <code class="docutils literal notranslate"><span class="pre">arr.item()</span></code> to retrieve the numerical entry from a 0D array:</p>
<div class="highlight-python notranslate"><div class="highlight"><pre><span></span><span class="gp">>>> </span><span class="n">x</span><span class="o">.</span><span class="n">item</span><span class="p">()</span>
<span class="go">15.2</span>
</pre></div>
</div>
<p>Zero-dimensional arrays do not show up in real applications very often. They are, however, important from the point of view of NumPy being self-consistent in how it treats dimensionality in its arrays, and it is important that you are at least exposed to a 0D array and understand its nuances.</p>
<div class="admonition note">
<p class="admonition-title fa fa-exclamation-circle"><strong>Takeaway</strong>:</p>
<p>Although accessing data along varying dimensions is ultimately all a matter of judicious bookkeeping (you <em>could</em> access all of this data from a 1-dimensional array, after all), NumPy’s ability to provide users with an interface for accessing data along dimensions is incredibly useful. It affords us an ability to impose intuitive, abstract structure to our data.</p>
</div>
</div>
<div class="section" id="Manipulating-Arrays">
<h2>Manipulating Arrays<a class="headerlink" href="#Manipulating-Arrays" title="Permalink to this headline"></a></h2>
<p>NumPy provides an assortment of functions that allow us manipulate the way that an array’s data can be accessed. These permit us to reshape an array, change its dimensionality, and swap the positions of its axes:</p>
<div class="highlight-python notranslate"><div class="highlight"><pre><span></span><span class="gp">>>> </span><span class="n">x</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">array</span><span class="p">([[</span> <span class="mi">1</span><span class="p">,</span> <span class="mi">2</span><span class="p">,</span> <span class="mi">3</span><span class="p">,</span> <span class="mi">4</span><span class="p">],</span>
<span class="gp">... </span> <span class="p">[</span> <span class="mi">5</span><span class="p">,</span> <span class="mi">6</span><span class="p">,</span> <span class="mi">7</span><span class="p">,</span> <span class="mi">8</span><span class="p">],</span>
<span class="gp">... </span> <span class="p">[</span> <span class="mi">9</span><span class="p">,</span> <span class="mi">10</span><span class="p">,</span> <span class="mi">11</span><span class="p">,</span> <span class="mi">12</span><span class="p">]])</span>
<span class="go"># reshaping an array</span>
<span class="gp">>>> </span><span class="n">x</span><span class="o">.</span><span class="n">reshape</span><span class="p">(</span><span class="mi">3</span><span class="p">,</span> <span class="mi">2</span><span class="p">,</span> <span class="mi">2</span><span class="p">)</span>
<span class="go">array([[[ 1, 2],</span>
<span class="go"> [ 3, 4]],</span>
<span class="go"> [[ 5, 6],</span>
<span class="go"> [ 7, 8]],</span>
<span class="go"> [[ 9, 10],</span>
<span class="go"> [11, 12]]])</span>
<span class="go"># Transposing an array: reversing</span>
<span class="go"># the ordering of its axes. This interchanges</span>
<span class="go"># the rows and columns of `x`</span>
<span class="gp">>>> </span><span class="n">x</span><span class="o">.</span><span class="n">transpose</span><span class="p">()</span>
<span class="go">array([[ 1, 5, 9],</span>
<span class="go"> [ 2, 6, 10],</span>
<span class="go"> [ 3, 7, 11],</span>
<span class="go"> [ 4, 8, 12]])</span>
</pre></div>
</div>
<p>A complete listing of the available array-manipulation functions can be found in the <a class="reference external" href="https://numpy.org/doc/stable/reference/routines.array-manipulation.html">official NumPy documentation</a>. Among these functions, the reshape function is especially useful.</p>
<div class="section" id="Introducing-the-reshape-Function">
<h3>Introducing the <code class="docutils literal notranslate"><span class="pre">reshape</span></code> Function<a class="headerlink" href="#Introducing-the-reshape-Function" title="Permalink to this headline"></a></h3>
<p>The <code class="docutils literal notranslate"><span class="pre">reshape</span></code> function allows you to change the dimensionality and axis-layout of a given array. This adjusts the indexing interface used to access the array’s underlying data, as was discussed in earlier in this module. Let’s take a shape-(6,) array, and reshape it to a shape-(2, 3) array:</p>
<div class="highlight-python notranslate"><div class="highlight"><pre><span></span><span class="gp">>>> </span><span class="kn">import</span> <span class="nn">numpy</span> <span class="k">as</span> <span class="nn">np</span>
<span class="gp">>>> </span><span class="n">x</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">array</span><span class="p">([</span><span class="mi">0</span><span class="p">,</span> <span class="mi">1</span><span class="p">,</span> <span class="mi">2</span><span class="p">,</span> <span class="mi">3</span><span class="p">,</span> <span class="mi">4</span><span class="p">,</span> <span class="mi">5</span><span class="p">])</span>
<span class="go"># reshape a shape-(6,) array into a shape-(2,3) array</span>
<span class="gp">>>> </span><span class="n">x</span><span class="o">.</span><span class="n">reshape</span><span class="p">(</span><span class="mi">2</span><span class="p">,</span> <span class="mi">3</span><span class="p">)</span>
<span class="go">array([[0, 1, 2],</span>
<span class="go"> [3, 4, 5]])</span>
</pre></div>
</div>
<p>You can also conveniently reshape an array by “setting” its shape via assignment:</p>
<div class="highlight-python notranslate"><div class="highlight"><pre><span></span><span class="c1"># equivalent to: x = x.reshape(2, 3)</span>
<span class="o">>>></span> <span class="n">x</span><span class="o">.</span><span class="n">shape</span> <span class="o">=</span> <span class="p">(</span><span class="mi">2</span><span class="p">,</span> <span class="mi">3</span><span class="p">)</span>
</pre></div>
</div>
<p>Of course, the size the the initial array must match the size of the to-be reshaped array:</p>
<div class="highlight-python notranslate"><div class="highlight"><pre><span></span><span class="c1"># an array with 5 numbers are cannot be reshaped</span>
<span class="c1"># into a (3, 2) array</span>
<span class="o">>>></span> <span class="n">np</span><span class="o">.</span><span class="n">array</span><span class="p">([</span><span class="mi">0</span><span class="p">,</span> <span class="mi">1</span><span class="p">,</span> <span class="mi">2</span><span class="p">,</span> <span class="mi">3</span><span class="p">,</span> <span class="mi">4</span><span class="p">])</span><span class="o">.</span><span class="n">reshape</span><span class="p">(</span><span class="mi">3</span><span class="p">,</span> <span class="mi">2</span><span class="p">)</span>
<span class="ne">ValueError</span><span class="p">:</span> <span class="n">total</span> <span class="n">size</span> <span class="n">of</span> <span class="n">new</span> <span class="n">array</span> <span class="n">must</span> <span class="n">be</span> <span class="n">unchanged</span>
</pre></div>
</div>
<p>Multidimensional arrays can be reshaped too:</p>
<div class="highlight-python notranslate"><div class="highlight"><pre><span></span><span class="c1"># reshaping a multidimensional array</span>
<span class="o">>>></span> <span class="n">x</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">array</span><span class="p">([[</span> <span class="mi">0</span><span class="p">,</span> <span class="mi">1</span><span class="p">,</span> <span class="mi">2</span><span class="p">,</span> <span class="mi">3</span><span class="p">],</span>
<span class="o">...</span> <span class="p">[</span> <span class="mi">4</span><span class="p">,</span> <span class="mi">5</span><span class="p">,</span> <span class="mi">6</span><span class="p">,</span> <span class="mi">7</span><span class="p">],</span>
<span class="o">...</span> <span class="p">[</span> <span class="mi">8</span><span class="p">,</span> <span class="mi">9</span><span class="p">,</span> <span class="mi">10</span><span class="p">,</span> <span class="mi">11</span><span class="p">]])</span>
<span class="c1"># reshape from (3, 4) to (2, 3, 2)</span>
<span class="o">>>></span> <span class="n">x</span><span class="o">.</span><span class="n">reshape</span><span class="p">(</span><span class="mi">2</span><span class="p">,</span> <span class="mi">3</span><span class="p">,</span> <span class="mi">2</span><span class="p">)</span>
<span class="n">array</span><span class="p">([[[</span> <span class="mi">0</span><span class="p">,</span> <span class="mi">1</span><span class="p">],</span>
<span class="p">[</span> <span class="mi">2</span><span class="p">,</span> <span class="mi">3</span><span class="p">],</span>
<span class="p">[</span> <span class="mi">4</span><span class="p">,</span> <span class="mi">5</span><span class="p">]],</span>
<span class="p">[[</span> <span class="mi">6</span><span class="p">,</span> <span class="mi">7</span><span class="p">],</span>
<span class="p">[</span> <span class="mi">8</span><span class="p">,</span> <span class="mi">9</span><span class="p">],</span>
<span class="p">[</span><span class="mi">10</span><span class="p">,</span> <span class="mi">11</span><span class="p">]]])</span>
</pre></div>
</div>
<p>Because the size of an input array and the resulting reshaped array must agree, you can specify <em>one</em> of the dimension-sizes in the reshape function to be -1, and this will cue NumPy to compute that dimension’s size for you. For example, if you are reshaping a shape-(36,) array into a shape-(3, 4, 3) array. The following are valid:</p>
<div class="highlight-python notranslate"><div class="highlight"><pre><span></span><span class="c1"># Equivalent ways of specifying a reshape</span>
<span class="c1"># np.arange(36) produces the shape-(36,) array ([0, 1, 2, ..., 35])</span>
<span class="n">np</span><span class="o">.</span><span class="n">arange</span><span class="p">(</span><span class="mi">36</span><span class="p">)</span><span class="o">.</span><span class="n">reshape</span><span class="p">(</span><span class="mi">3</span><span class="p">,</span> <span class="mi">4</span><span class="p">,</span> <span class="mi">3</span><span class="p">)</span> <span class="c1"># (36,) --reshape--> (3, 4, 3)</span>
<span class="n">np</span><span class="o">.</span><span class="n">arange</span><span class="p">(</span><span class="mi">36</span><span class="p">)</span><span class="o">.</span><span class="n">reshape</span><span class="p">(</span><span class="mi">3</span><span class="p">,</span> <span class="mi">4</span><span class="p">,</span> <span class="o">-</span><span class="mi">1</span><span class="p">)</span> <span class="c1"># NumPy replaces -1 with 36/(3*4) -> 3</span>
<span class="n">np</span><span class="o">.</span><span class="n">arange</span><span class="p">(</span><span class="mi">36</span><span class="p">)</span><span class="o">.</span><span class="n">reshape</span><span class="p">(</span><span class="mi">3</span><span class="p">,</span> <span class="o">-</span><span class="mi">1</span><span class="p">,</span> <span class="mi">3</span><span class="p">)</span> <span class="c1"># NumPy replaces -1 with 36/(3*3) -> 4</span>
<span class="n">np</span><span class="o">.</span><span class="n">arange</span><span class="p">(</span><span class="mi">36</span><span class="p">)</span><span class="o">.</span><span class="n">reshape</span><span class="p">(</span><span class="o">-</span><span class="mi">1</span><span class="p">,</span> <span class="mi">4</span><span class="p">,</span> <span class="mi">3</span><span class="p">)</span> <span class="c1"># NumPy replaces -1 with 36/(3*4) -> 3</span>
</pre></div>
</div>
<p>You can use -1 to specify only one dimension:</p>
<div class="highlight-python notranslate"><div class="highlight"><pre><span></span><span class="gp">>>> </span><span class="n">np</span><span class="o">.</span><span class="n">arange</span><span class="p">(</span><span class="mi">36</span><span class="p">)</span><span class="o">.</span><span class="n">reshape</span><span class="p">(</span><span class="mi">3</span><span class="p">,</span> <span class="o">-</span><span class="mi">1</span><span class="p">,</span> <span class="o">-</span><span class="mi">1</span><span class="p">)</span> <span class="c1"># this is an ambiguous specification, and thus</span>
<span class="go">---------------------------------------------------------------------------</span>
<span class="go">ValueError Traceback (most recent call last)</span>
<span class="go"><ipython-input-3-207d18d18af2> in <module>()</span>
<span class="go">----> 1 np.arange(36).reshape(3, -1, -1)</span>
<span class="go">ValueError: can only specify one unknown dimension</span>
</pre></div>
</div>
<div class="admonition note">
<p class="admonition-title fa fa-exclamation-circle"><strong>Reshaping Does Not Make a Copy of an Array</strong>:</p>
<p>For all straightforward applications of reshape, NumPy does not actually create a new copy of an array’s data when performing a <code class="docutils literal notranslate"><span class="pre">reshape</span></code> operation. Instead, the original array and the reshaped array reference the same underlying data. The reshaped array simply provides a new index-interface for accessing said data, and is thus referred to as a “view” of the original array (more on this “views” in a later section).</p>
</div>
</div>
</div>
<div class="section" id="Links-to-Official-Documentation">
<h2>Links to Official Documentation<a class="headerlink" href="#Links-to-Official-Documentation" title="Permalink to this headline"></a></h2>
<ul class="simple">
<li><p><a class="reference external" href="https://numpy.org/doc/stable/reference/arrays.ndarray.html">The N-dimensional array</a></p></li>
<li><p><a class="reference external" href="https://numpy.org/doc/stable/user/basics.indexing.html#indexing">Array indexing</a></p></li>
<li><p><a class="reference external" href="https://numpy.org/doc/stable/reference/routines.indexing.html#indexing-routines">Indexing routines</a></p></li>
<li><p><a class="reference external" href="https://numpy.org/doc/stable/reference/routines.array-manipulation.html">Array manipulation routines</a></p></li>
</ul>
</div>
<div class="section" id="Reading-Comprehension-Solutions">
<h2>Reading Comprehension Solutions<a class="headerlink" href="#Reading-Comprehension-Solutions" title="Permalink to this headline"></a></h2>
<p><strong>Reading Comprehension: Multi-dimensional Indexing</strong></p>
<div class="highlight-python notranslate"><div class="highlight"><pre><span></span><span class="gp">>>> </span><span class="n">arr</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">array</span><span class="p">([[[</span> <span class="mi">0</span><span class="p">,</span> <span class="mi">1</span><span class="p">,</span> <span class="mi">2</span><span class="p">],</span>
<span class="gp">... </span> <span class="p">[</span> <span class="mi">3</span><span class="p">,</span> <span class="mi">4</span><span class="p">,</span> <span class="mi">5</span><span class="p">],</span>
<span class="gp">... </span> <span class="p">[</span> <span class="mi">6</span><span class="p">,</span> <span class="mi">7</span><span class="p">,</span> <span class="mi">8</span><span class="p">]],</span>
<span class="gp">...</span>
<span class="gp">... </span> <span class="p">[[</span> <span class="mi">9</span><span class="p">,</span> <span class="mi">10</span><span class="p">,</span> <span class="mi">11</span><span class="p">],</span>
<span class="gp">... </span> <span class="p">[</span><span class="mi">12</span><span class="p">,</span> <span class="mi">13</span><span class="p">,</span> <span class="mi">14</span><span class="p">],</span>
<span class="gp">... </span> <span class="p">[</span><span class="mi">15</span><span class="p">,</span> <span class="mi">16</span><span class="p">,</span> <span class="mi">17</span><span class="p">]],</span>
<span class="gp">...</span>
<span class="gp">... </span> <span class="p">[[</span><span class="mi">18</span><span class="p">,</span> <span class="mi">19</span><span class="p">,</span> <span class="mi">20</span><span class="p">],</span>
<span class="gp">... </span> <span class="p">[</span><span class="mi">21</span><span class="p">,</span> <span class="mi">22</span><span class="p">,</span> <span class="mi">23</span><span class="p">],</span>
<span class="gp">... </span> <span class="p">[</span><span class="mi">24</span><span class="p">,</span> <span class="mi">25</span><span class="p">,</span> <span class="mi">26</span><span class="p">]]])</span>
</pre></div>
</div>
<div class="highlight-python notranslate"><div class="highlight"><pre><span></span><span class="c1">#1</span>
<span class="o">>>></span> <span class="n">arr</span><span class="p">[:,</span> <span class="p">:,</span> <span class="mi">2</span><span class="p">]</span>
<span class="n">array</span><span class="p">([[</span> <span class="mi">2</span><span class="p">,</span> <span class="mi">5</span><span class="p">,</span> <span class="mi">8</span><span class="p">],</span>
<span class="p">[</span><span class="mi">11</span><span class="p">,</span> <span class="mi">14</span><span class="p">,</span> <span class="mi">17</span><span class="p">],</span>
<span class="p">[</span><span class="mi">20</span><span class="p">,</span> <span class="mi">23</span><span class="p">,</span> <span class="mi">26</span><span class="p">]])</span>
<span class="c1">#2</span>
<span class="o">>>></span> <span class="n">arr</span><span class="p">[</span><span class="mi">0</span><span class="p">:</span><span class="mi">2</span><span class="p">,</span> <span class="mi">1</span><span class="p">,</span> <span class="p">:]</span>
<span class="n">array</span><span class="p">([[</span> <span class="mi">3</span><span class="p">,</span> <span class="mi">4</span><span class="p">,</span> <span class="mi">5</span><span class="p">],</span>
<span class="p">[</span><span class="mi">12</span><span class="p">,</span> <span class="mi">13</span><span class="p">,</span> <span class="mi">14</span><span class="p">]])</span>
<span class="c1">#3</span>
<span class="o">>>></span> <span class="n">arr</span><span class="p">[</span><span class="mi">0</span><span class="p">,</span> <span class="p">:</span><span class="mi">2</span><span class="p">,</span> <span class="mi">2</span><span class="p">]</span>
<span class="n">array</span><span class="p">([</span><span class="mi">2</span><span class="p">,</span> <span class="mi">5</span><span class="p">])</span>
<span class="c1">#4</span>
<span class="o">>>></span> <span class="n">arr</span><span class="p">[</span><span class="mi">1</span><span class="p">,</span> <span class="p">:,</span> <span class="p">::</span><span class="o">-</span><span class="mi">1</span><span class="p">]</span>
<span class="n">array</span><span class="p">([[</span><span class="mi">11</span><span class="p">,</span> <span class="mi">10</span><span class="p">,</span> <span class="mi">9</span><span class="p">],</span>
<span class="p">[</span><span class="mi">14</span><span class="p">,</span> <span class="mi">13</span><span class="p">,</span> <span class="mi">12</span><span class="p">],</span>
<span class="p">[</span><span class="mi">17</span><span class="p">,</span> <span class="mi">16</span><span class="p">,</span> <span class="mi">15</span><span class="p">]])</span>
</pre></div>
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