1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
27
28
29
30
31
32
33
34
35
36
37
38
39
40
41
42
43
44
45
46
47
48
49
50
51
52
53
54
55
56
57
58
59
60
61
62
63
64
65
66
67
68
69
70
71
72
73
74
75
76
77
78
79
80
81
82
83
84
85
86
87
88
89
90
91
92
93
94
95
96
97
98
99
100
101
102
103
104
105
106
107
108
109
110
111
112
113
114
115
116
117
118
119
120
121
122
123
124
125
126
127
128
129
130
131
132
133
134
135
136
137
138
139
140
141
142
143
144
145
146
147
148
149
150
151
152
153
154
155
156
157
158
159
160
161
162
163
164
165
166
167
168
169
|
<head>
<link href="../favicon.ico" rel="shortcut icon" type="image/x-icon"/>
<link href="../style.css" rel="stylesheet"/>
<title>BQN: Cells and Rank</title>
</head>
<div class="nav">(<a href="https://github.com/mlochbaum/BQN">github</a>) / <a href="../index.html">BQN</a> / <a href="index.html">doc</a></div>
<h1 id="cells-and-rank"><a class="header" href="#cells-and-rank">Cells and Rank</a></h1>
<p>The Cells modifier <code><span class='Modifier'>˘</span></code> applies a function to major cells of its argument, much like <a href="map.html">Each</a> applies to elements. Each result from <code><span class='Function'>𝔽</span></code> becomes a major cell of the result, which means they must all have the same shape.</p>
<p>The Rank modifier <code><span class='Modifier2'>⎉</span></code> generalizes this concept by allowing numbers provided by <code><span class='Function'>𝔾</span></code> to specify a rank for each argument: non-negative to indicate the rank of each array passed to <code><span class='Function'>𝔽</span></code>, or negative for the number of axes that are mapped over. Cells, which maps over one axis of each argument, is identical to <code><span class='Modifier2'>⎉</span><span class='Number'>¯1</span></code>. Rank is analogous to the <a href="depth.html#the-depth-modifier">Depth modifier</a>, but the homogeneous structure of an array eliminates some tricky edge cases found in Depth.</p>
<h2 id="cells"><a class="header" href="#cells">Cells</a></h2>
<p>The function Cells (<code><span class='Modifier'>˘</span></code>) is named after <em>major cells</em> in an array. A major cell is a component of an array with dimension one smaller, so that the major cells of a list are <a href="enclose.html#whats-a-unit">units</a>, the major cells of a rank-2 table are its rows (which are lists), and the major cells of a rank-3 array are tables.</p>
<p>The function <code><span class='Function'>𝔽</span><span class='Modifier'>˘</span></code> applies <code><span class='Function'>𝔽</span></code> to the major cells of <code><span class='Value'>𝕩</span></code>. So, for example, where <a href="shift.html">Nudge</a> (<code><span class='Function'>»</span></code>) shifts an entire table, Nudge Cells shifts its major cells, or rows.</p>
<a class="replLink" title="Open in the REPL" target="_blank" href="https://mlochbaum.github.io/BQN/try.html#code=YSDihpAgJ2EnICsgM+KAv+KImCDipYog4oaVMjQgICMgQSBjaGFyYWN0ZXIgYXJyYXkKCuKfqCAgYSAgICAgICwgICAgIMK7YSAgICAgLCAgICDCu8uYYSDin6k=">↗️</a><pre> <span class='Value'>a</span> <span class='Gets'>←</span> <span class='String'>'a'</span> <span class='Function'>+</span> <span class='Number'>3</span><span class='Ligature'>‿</span><span class='Modifier2'>∘</span> <span class='Function'>⥊</span> <span class='Function'>↕</span><span class='Number'>24</span> <span class='Comment'># A character array
</span>
<span class='Bracket'>⟨</span> <span class='Value'>a</span> <span class='Separator'>,</span> <span class='Function'>»</span><span class='Value'>a</span> <span class='Separator'>,</span> <span class='Function'>»</span><span class='Modifier'>˘</span><span class='Value'>a</span> <span class='Bracket'>⟩</span>
┌─
· ┌─ ┌─ ┌─
╵"abcdefgh ╵" ╵" abcdefg
ijklmnop abcdefgh ijklmno
qrstuvwx" ijklmnop" qrstuvw"
┘ ┘ ┘
┘
</pre>
<p>What's it mean for Nudge to shift the "entire table"? The block above shows that it shifts downward, but what's really happening is that Nudge treats <code><span class='Value'>𝕩</span></code> as a collection of major cells—its rows—and shifts these. So it adds an entire row and moves the rest of the rows downwards. Nudge Cells appears similar, but it's acting independently on each row, and the values that it moves around are major cells of the row, that is, rank-0 units.</p>
<p>Here's an example showing how Cells can be used to shift each row independently, even though it's not possible to shift columns like this (in fact the best way to do that would be to <a href="transpose.html">transpose</a> in order to work on rows). It uses the not-yet-introduced dyadic form of Cells, so you might want to come back to it after reading the next section.</p>
<a class="replLink" title="Open in the REPL" target="_blank" href="https://mlochbaum.github.io/BQN/try.html#code=KOKGkSLiiJjiiJgiKSDiipHiirjCu8uYIGE=">↗️</a><pre> <span class='Paren'>(</span><span class='Function'>↑</span><span class='String'>"∘∘"</span><span class='Paren'>)</span> <span class='Function'>⊑</span><span class='Modifier2'>⊸</span><span class='Function'>»</span><span class='Modifier'>˘</span> <span class='Value'>a</span>
┌─
╵"abcdefgh
∘ijklmno
∘∘qrstuv"
┘
</pre>
<p>You can also see how Cells splits its argument into rows using a less array-oriented primitive: <a href="enclose.html">Enclose</a> just wraps each row up so that it appears as a separate element in the final result.</p>
<a class="replLink" title="Open in the REPL" target="_blank" href="https://mlochbaum.github.io/BQN/try.html#code=PMuYIGE=">↗️</a><pre> <span class='Function'><</span><span class='Modifier'>˘</span> <span class='Value'>a</span>
⟨ "abcdefgh" "ijklmnop" "qrstuvwx" ⟩
</pre>
<p>Enclose also comes in handy for the following task: join the rows in an array of lists, resulting in an array where each element is a joined row. The obvious guess would be "join cells", <code><span class='Function'>∾</span><span class='Modifier'>˘</span></code>, but it doesn't work, because each <code><span class='Function'>∾</span></code> can return a result with a different length. Cells tries to make each result of <code><span class='Function'>∾</span></code> into a <em>cell</em>, when the problem was to use it as an <em>element</em>. But a 0-cell is an enclosed element, so we can close the gap by applying <code><span class='Function'><</span></code> to a joined list: <code><span class='Function'><</span><span class='Modifier2'>∘</span><span class='Function'>∾</span></code>.</p>
<a class="replLink" title="Open in the REPL" target="_blank" href="https://mlochbaum.github.io/BQN/try.html#code=4oqiIHMg4oaQICJ3b3JkcyLigL8iZ28i4oC/ImhlcmUiIOKJjSAic29tZSLigL8ib3RoZXIi4oC/IndvcmRzIgoK4oi+y5ggcwoKPOKImOKIvsuYIHM=">↗️</a><pre> <span class='Function'>⊢</span> <span class='Value'>s</span> <span class='Gets'>←</span> <span class='String'>"words"</span><span class='Ligature'>‿</span><span class='String'>"go"</span><span class='Ligature'>‿</span><span class='String'>"here"</span> <span class='Function'>≍</span> <span class='String'>"some"</span><span class='Ligature'>‿</span><span class='String'>"other"</span><span class='Ligature'>‿</span><span class='String'>"words"</span>
┌─
╵ "words" "go" "here"
"some" "other" "words"
┘
<span class='Function'>∾</span><span class='Modifier'>˘</span> <span class='Value'>s</span>
<span class='Error'>Error: >: Elements didn't have equal shapes (contained ⟨11⟩ and ⟨14⟩)</span>
<span class='Function'><</span><span class='Modifier2'>∘</span><span class='Function'>∾</span><span class='Modifier'>˘</span> <span class='Value'>s</span>
⟨ "wordsgohere" "someotherwords" ⟩
</pre>
<p>This approach can apply to more complicated functions as well. And because the result of <code><span class='Function'><</span></code> always has the same shape, <code><span class='Bracket'>⟨⟩</span></code>, the function <code><span class='Function'><</span><span class='Modifier2'>∘</span><span class='Function'>𝔽</span><span class='Modifier'>˘</span></code> can never have a shape agreement error. So if <code><span class='Function'>𝔽</span><span class='Modifier'>˘</span></code> fails, it can't hurt to check <code><span class='Function'><</span><span class='Modifier2'>∘</span><span class='Function'>𝔽</span><span class='Modifier'>˘</span></code> and see what results <code><span class='Function'>𝔽</span></code> is returning.</p>
<h3 id="two-arguments"><a class="header" href="#two-arguments">Two arguments</a></h3>
<p>When given two arguments, Cells tries to pair their cells together. Starting simple, a unit (whether array or atom) on either side will be paired with every cell of the other argument.</p>
<a class="replLink" title="Open in the REPL" target="_blank" href="https://mlochbaum.github.io/BQN/try.html#code=J+KImCcgwrvLmCBh">↗️</a><pre> <span class='String'>'∘'</span> <span class='Function'>»</span><span class='Modifier'>˘</span> <span class='Value'>a</span>
┌─
╵"∘abcdefg
∘ijklmno
∘qrstuvw"
┘
</pre>
<p>If you <em>want</em> to use this one-to-many behavior with an array, it'll take more work: since you're really only mapping over one argument, <a href="hook.html">bind</a> the other inside Cells.</p>
<a class="replLink" title="Open in the REPL" target="_blank" href="https://mlochbaum.github.io/BQN/try.html#code=IuKImOKImCIgwrvLmCBhCgoi4oiY4oiYIuKKuMK7y5ggYQ==">↗️</a><pre> <span class='String'>"∘∘"</span> <span class='Function'>»</span><span class='Modifier'>˘</span> <span class='Value'>a</span>
<span class='Error'>Error: ˘: Leading axis of arguments not equal (⟨2⟩ ≡ ≢𝕨, 3‿8 ≡ ≢𝕩)</span>
<span class='String'>"∘∘"</span><span class='Modifier2'>⊸</span><span class='Function'>»</span><span class='Modifier'>˘</span> <span class='Value'>a</span>
┌─
╵"∘∘abcdef
∘∘ijklmn
∘∘qrstuv"
┘
</pre>
<p>This is because the general case of Cells does one-to-one matching, pairing the first axis of one argument with the other. For this to work, the two arguments need to have the same length.</p>
<a class="replLink" title="Open in the REPL" target="_blank" href="https://mlochbaum.github.io/BQN/try.html#code=4p+oICIwMTIiIMK7y5ggYSwgICgz4oC/4oiY4qWKIlVWV1hZWiIpIMK7y5ggYSDin6k=">↗️</a><pre> <span class='Bracket'>⟨</span> <span class='String'>"012"</span> <span class='Function'>»</span><span class='Modifier'>˘</span> <span class='Value'>a</span><span class='Separator'>,</span> <span class='Paren'>(</span><span class='Number'>3</span><span class='Ligature'>‿</span><span class='Modifier2'>∘</span><span class='Function'>⥊</span><span class='String'>"UVWXYZ"</span><span class='Paren'>)</span> <span class='Function'>»</span><span class='Modifier'>˘</span> <span class='Value'>a</span> <span class='Bracket'>⟩</span>
┌─
· ┌─ ┌─
╵"0abcdefg ╵"UVabcdef
1ijklmno WXijklmn
2qrstuvw" YZqrstuv"
┘ ┘
┘
</pre>
<p>The arguments might have different ranks: for example, <code><span class='String'>"012"</span></code> has rank 1 and <code><span class='Value'>a</span></code> has rank 2 above. That's fine: it just means Cells will pass arguments of rank 0 and 1 to its operand. You can see these arguments using <a href="pair.html">Pair</a> Cells, <code><span class='Function'>⋈</span><span class='Modifier'>˘</span></code>, so that each cell of the result is just a list of the two arguments used for that call.</p>
<a class="replLink" title="Open in the REPL" target="_blank" href="https://mlochbaum.github.io/BQN/try.html#code=IjAxMiIg4ouIy5ggYQ==">↗️</a><pre> <span class='String'>"012"</span> <span class='Function'>⋈</span><span class='Modifier'>˘</span> <span class='Value'>a</span>
┌─
╵ ┌· "abcdefgh"
·'0'
┘
┌· "ijklmnop"
·'1'
┘
┌· "qrstuvwx"
·'2'
┘
┘
</pre>
<h2 id="rank"><a class="header" href="#rank">Rank</a></h2>
<p>Rank (<code><span class='Modifier2'>⎉</span></code>) is a generalization of Cells (<code><span class='Function'>𝔽</span><span class='Modifier'>˘</span></code> is defined to be <code><span class='Function'>𝔽</span><span class='Modifier2'>⎉</span><span class='Number'>¯1</span></code>) that can apply to arbitrary—not just major—cells and combine different levels of mapping for two arguments.</p>
<p>Rank comes in handy when there are high-rank arrays with lots of exciting axes, which is a great use case for BQN but honestly isn't all that common. And to continue this trend of honesty, using Rank just never <em>feels</em> good—it's some heavy machinery that I drag out when nothing else works, only to make use of a small part of the functionality. If Cells covers your use cases, that's probably for the best!</p>
<h3 id="negative-and-positive-ranks"><a class="header" href="#negative-and-positive-ranks">Negative and positive ranks</a></h3>
<p>I've said that <code><span class='Function'>𝔽</span><span class='Modifier2'>⎉</span><span class='Number'>¯1</span></code> is <code><span class='Function'>𝔽</span><span class='Modifier'>˘</span></code>. And it's also the case that <code><span class='Function'>𝔽</span><span class='Modifier2'>⎉</span><span class='Number'>¯2</span></code> is <code><span class='Function'>𝔽</span><span class='Modifier'>˘˘</span></code>. And <code><span class='Function'>𝔽</span><span class='Modifier2'>⎉</span><span class='Number'>¯3</span></code> is <code><span class='Function'>𝔽</span><span class='Modifier'>˘˘˘</span></code>. And so on.</p>
<a class="replLink" title="Open in the REPL" target="_blank" href="https://mlochbaum.github.io/BQN/try.html#code=KOKGlTQpICjii4jLmMuYy5gg4omhIOKLiOKOicKvMykg4oaVNOKAvzLigL8y4oC/NQ==">↗️</a><pre> <span class='Paren'>(</span><span class='Function'>↕</span><span class='Number'>4</span><span class='Paren'>)</span> <span class='Paren'>(</span><span class='Function'>⋈</span><span class='Modifier'>˘˘˘</span> <span class='Function'>≡</span> <span class='Function'>⋈</span><span class='Modifier2'>⎉</span><span class='Number'>¯3</span><span class='Paren'>)</span> <span class='Function'>↕</span><span class='Number'>4</span><span class='Ligature'>‿</span><span class='Number'>2</span><span class='Ligature'>‿</span><span class='Number'>2</span><span class='Ligature'>‿</span><span class='Number'>5</span>
1
</pre>
<p>So <code><span class='Function'>𝔽</span><span class='Modifier2'>⎉</span><span class='Paren'>(</span><span class='Function'>-</span><span class='Value'>k</span><span class='Paren'>)</span></code>, at least for <code><span class='Value'>k</span><span class='Function'>≥</span><span class='Number'>1</span></code>, is how you map over the first <code><span class='Value'>k</span></code> axes of an array or two. We'll get more into why this is in the next section. What about some positivity for a change?</p>
<a class="replLink" title="Open in the REPL" target="_blank" href="https://mlochbaum.github.io/BQN/try.html#code=POKOiTAgImFiYyLiiY0iZGVmIgoKPOKOiTEgImFiYyLiiY0iZGVmIg==">↗️</a><pre> <span class='Function'><</span><span class='Modifier2'>⎉</span><span class='Number'>0</span> <span class='String'>"abc"</span><span class='Function'>≍</span><span class='String'>"def"</span>
┌─
╵ ┌· ┌· ┌·
·'a' ·'b' ·'c'
┘ ┘ ┘
┌· ┌· ┌·
·'d' ·'e' ·'f'
┘ ┘ ┘
┘
<span class='Function'><</span><span class='Modifier2'>⎉</span><span class='Number'>1</span> <span class='String'>"abc"</span><span class='Function'>≍</span><span class='String'>"def"</span>
⟨ "abc" "def" ⟩
</pre>
<p>The function <code><span class='Function'>𝔽</span><span class='Modifier2'>⎉</span><span class='Value'>k</span></code>, for <code><span class='Value'>k</span><span class='Function'>≥</span><span class='Number'>0</span></code>, operates on the <code><span class='Value'>k</span></code>-cells of its arguments—that is, it maps over all <em>but</em> the last <code><span class='Value'>k</span></code> axes. For any given argument <code><span class='Value'>a</span></code>, ranks <code><span class='Value'>k</span></code> and <code><span class='Value'>k</span><span class='Function'>-=</span><span class='Value'>a</span></code> are the same, as long as <code><span class='Value'>k</span><span class='Function'>≥</span><span class='Number'>0</span></code> and <code><span class='Paren'>(</span><span class='Value'>k</span><span class='Function'>-=</span><span class='Value'>a</span><span class='Paren'>)</span><span class='Function'>≤</span><span class='Number'>¯1</span></code>. So rank 2 is rank ¯3 for a rank-5 array. The reason this option is useful is that the same rank might be applied to multiple arguments, either with multiple function calls or one call on two arguments. Let's revisit an example with Cells from before, shifting the same string into each row of a table. The function <code><span class='Function'>»</span></code> should be called on rank-1 strings, but because the argument ranks are different, a negative rank can't get down to rank 1 on both sides. Positive rank 1 does the job, allowing us to unbundle the string <code><span class='String'>"∘∘"</span></code> so that <code><span class='Function'>»</span><span class='Modifier2'>⎉</span><span class='Number'>1</span></code> is a standalone function.</p>
<a class="replLink" title="Open in the REPL" target="_blank" href="https://mlochbaum.github.io/BQN/try.html#code=IuKImOKImCLiirjCu8uYIGEKCiLiiJjiiJgiIMK74o6JMSBh">↗️</a><pre> <span class='String'>"∘∘"</span><span class='Modifier2'>⊸</span><span class='Function'>»</span><span class='Modifier'>˘</span> <span class='Value'>a</span>
┌─
╵"∘∘abcdef
∘∘ijklmn
∘∘qrstuv"
┘
<span class='String'>"∘∘"</span> <span class='Function'>»</span><span class='Modifier2'>⎉</span><span class='Number'>1</span> <span class='Value'>a</span>
┌─
╵"∘∘abcdef
∘∘ijklmn
∘∘qrstuv"
┘
</pre>
<p>The rank for a given argument is clamped, so that on a rank 3 argument for example, a rank of ¯5 counts as ¯3 and a rank of 6 counts as 3 (same for any other value less than ¯3 or greater than 3, although it does have to be a whole number). You may have noticed there's <a href="../commentary/problems.html#rankdepth-negative-zero">no</a> option for ¯0, "don't map over anything", but ∞ serves that purpose as it indicates the highest possible rank and thus the entire array. More on why that's useful later.</p>
<h3 id="frame-and-cells"><a class="header" href="#frame-and-cells">Frame and Cells</a></h3>
<h3 id="multiple-and-computed-ranks"><a class="header" href="#multiple-and-computed-ranks">Multiple and computed ranks</a></h3>
<p>The Rank modifier also accepts a list of one to three numbers for <code><span class='Value'>𝕘</span></code>, as well as a function <code><span class='Function'>𝔾</span></code> returning such a list. Practically speaking, here's what you need to know:</p>
<ul>
<li>A single number or one-element list indicates the ranks for all arguments.</li>
<li>Two numbers indicate the ranks for <code><span class='Value'>𝕨</span></code> and <code><span class='Value'>𝕩</span></code>.</li>
</ul>
<a class="replLink" title="Open in the REPL" target="_blank" href="https://mlochbaum.github.io/BQN/try.html#code=4oqiIG0g4oaQID7in6gw4oC/MeKAvzAswq8x4oC/MOKAvzAsMOKAvzDigL8x4p+pCgptICvLneKImMOX4o6JMeKAv+KIniAx4oC/MuKAvzMKCm0gK8ud4oiYw5fijokx4oC/4oieIDHigL8y4oC/M8OX4oycMeKAvzEwCgooImFiYyLiiY0iZGVmIikg4oi+4o6JMeKOiTHigL/iiJ4gPiJRUiLigL8iU1Qi4oC/IlVWIg==">↗️</a><pre> <span class='Function'>⊢</span> <span class='Value'>m</span> <span class='Gets'>←</span> <span class='Function'>></span><span class='Bracket'>⟨</span><span class='Number'>0</span><span class='Ligature'>‿</span><span class='Number'>1</span><span class='Ligature'>‿</span><span class='Number'>0</span><span class='Separator'>,</span><span class='Number'>¯1</span><span class='Ligature'>‿</span><span class='Number'>0</span><span class='Ligature'>‿</span><span class='Number'>0</span><span class='Separator'>,</span><span class='Number'>0</span><span class='Ligature'>‿</span><span class='Number'>0</span><span class='Ligature'>‿</span><span class='Number'>1</span><span class='Bracket'>⟩</span>
┌─
╵ 0 1 0
¯1 0 0
0 0 1
┘
<span class='Value'>m</span> <span class='Function'>+</span><span class='Modifier'>˝</span><span class='Modifier2'>∘</span><span class='Function'>×</span><span class='Modifier2'>⎉</span><span class='Number'>1</span><span class='Ligature'>‿</span><span class='Number'>∞</span> <span class='Number'>1</span><span class='Ligature'>‿</span><span class='Number'>2</span><span class='Ligature'>‿</span><span class='Number'>3</span>
⟨ 2 ¯1 3 ⟩
<span class='Value'>m</span> <span class='Function'>+</span><span class='Modifier'>˝</span><span class='Modifier2'>∘</span><span class='Function'>×</span><span class='Modifier2'>⎉</span><span class='Number'>1</span><span class='Ligature'>‿</span><span class='Number'>∞</span> <span class='Number'>1</span><span class='Ligature'>‿</span><span class='Number'>2</span><span class='Ligature'>‿</span><span class='Number'>3</span><span class='Function'>×</span><span class='Modifier'>⌜</span><span class='Number'>1</span><span class='Ligature'>‿</span><span class='Number'>10</span>
┌─
╵ 2 20
¯1 ¯10
3 30
┘
<span class='Paren'>(</span><span class='String'>"abc"</span><span class='Function'>≍</span><span class='String'>"def"</span><span class='Paren'>)</span> <span class='Function'>∾</span><span class='Modifier2'>⎉</span><span class='Number'>1</span><span class='Modifier2'>⎉</span><span class='Number'>1</span><span class='Ligature'>‿</span><span class='Number'>∞</span> <span class='Function'>></span><span class='String'>"QR"</span><span class='Ligature'>‿</span><span class='String'>"ST"</span><span class='Ligature'>‿</span><span class='String'>"UV"</span>
┌─
╎"abcQR
abcST
abcUV
·defQR
defST
defUV"
┘
</pre>
<p>Here's the full, boring description of how <code><span class='Function'>𝔾</span></code> is handled. The operand <code><span class='Function'>𝔾</span></code> is called on the arguments <code><span class='Value'>𝕨</span><span class='Function'>𝔾</span><span class='Value'>𝕩</span></code> before doing anything else (if it's not a function, this just returns <code><span class='Value'>𝕘</span></code>). Then it's converted to a list. It's required to have rank 0 or 1, but numbers and enclosed numbers are fine. This list can have one to three elements; three elements is the general case, as the elements give the ranks for monadic <code><span class='Value'>𝕩</span></code>, dyadic <code><span class='Value'>𝕨</span></code>, and dyadic <code><span class='Value'>𝕩</span></code> in order. If there are less than three elements, the list <code><span class='Value'>r</span></code> is expanded backwards-cyclically to <code><span class='Number'>3</span><span class='Modifier2'>⊸</span><span class='Function'>⥊</span><span class='Modifier2'>⌾</span><span class='Function'>⌽</span><span class='Value'>r</span></code>, turning <code><span class='Bracket'>⟨</span><span class='Value'>a</span><span class='Bracket'>⟩</span></code> into <code><span class='Value'>a</span><span class='Ligature'>‿</span><span class='Value'>a</span><span class='Ligature'>‿</span><span class='Value'>a</span></code> and <code><span class='Value'>a</span><span class='Ligature'>‿</span><span class='Value'>b</span></code> into <code><span class='Value'>b</span><span class='Ligature'>‿</span><span class='Value'>a</span><span class='Ligature'>‿</span><span class='Value'>b</span></code>. So <code><span class='Number'>3</span><span class='Modifier2'>⊸</span><span class='Function'>⥊</span><span class='Modifier2'>⌾</span><span class='Function'>⌽⥊</span><span class='Value'>𝕨</span><span class='Function'>𝔾</span><span class='Value'>𝕩</span></code> is the final formula.</p>
<h3 id="leading-axis-agreement"><a class="header" href="#leading-axis-agreement">Leading axis agreement</a></h3>
|
