lancejpollard/math-font-details.md

## math-font-details.md

      
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              math-font-details.md
            
          
    Math Font Checklist

Here's a concise checklist for font designers who want all these stretchy math constructs (parentheses, brackets, braces, integrals, summations, etc.) to render beautifully and correctly in KaTeX, MathJax, TeX, or any math-aware engine.
1. Understand what's going on


Math layout engines assemble tall symbols (like ⎛⎜⎝) by stacking multiple glyph parts, not by scaling one shape.
Each symbol family (parentheses, brackets, braces, integrals, summations) has top, middle/extender, and bottom pieces.
Modern math fonts encode their geometry and stacking rules in an OpenType MATH table.

2. Required glyphs

Include all the segment glyphs KaTeX and TeX expect:

Parentheses: ⎛ ⎜ ⎝ ⎞ ⎟ ⎠
Square brackets: ⎡ ⎢ ⎣ ⎤ ⎥ ⎦
Curly braces: ⎧ ⎨ ⎩ ⎫ ⎬ ⎭
Integrals and sums: ⌠ ⌡ ⎲ ⎳ ⎰ ⎱
Bars and ceilings/floors: ⎪ ⎮ ⌈ ⌉ ⌊ ⌋

Check that each one:

Aligns vertically on the baseline grid,
Has consistent stroke thickness and curvature,
Touches or slightly overlaps when stacked.

3. Define clean metrics


Set bounding boxes tightly around visible outlines (no excess vertical padding).
Keep consistent side bearings so pairs like ⎛⎞ visually balance.
Match vertical advance height across top/middle/bottom parts for perfect joins.
Ensure vertical alignment for bars (⎪, ⎮) and extenders.

4. MATH table essentials

If you want full TeX-style quality (beyond KaTeX's heuristics):
Add an OpenType MATH table specifying:

MathVariants: which glyphs form stretch assemblies (e.g. ( → top=⎛, mid=⎜, bottom=⎝).
MathConstants: axis height, rule thickness, fraction gaps, etc.
Italic corrections for slanted symbols.

You can build or inspect this using FontTools:
ttx -t MATH MyFont.otf
or reference STIX Two Math or Latin Modern Math as a model.
5. Test visually

Use:


KaTeX demo: https://katex.org


MathJax demo: https://mathjax.github.io/


hb-view (HarfBuzz) to test vertical joining.


Your font in:
.katex { font-family: "MyMathMono"; }


Render tall structures like:
\left( \begin{matrix}a\\b\\c\\d\\e\\f\\g\\h\\end{matrix} \right)
and verify seamless joins, balanced spacing, and good line alignment.
6. If your font is monospaced


It's fine for code, but monospacing breaks math proportions.
Narrow symbols (parentheses, brackets) will appear too far apart.
You can mitigate this by using smaller horizontal side bearings or variable width exceptions for math glyphs.

In short


Design top/mid/bottom pieces to connect cleanly, define tight metrics, include a proper MATH table, and visually test tall stacks in KaTeX/TeX.
Smooth joins + balanced spacing = beautiful stretchy symbols.

Quick yet Detailed Reference

Here is a complete yet concise reference for what a font designer must include so that all tall math delimiters and operators (parentheses, brackets, braces, integrals, sums) render correctly in KaTeX, MathJax, or TeX.
Everything below is self-contained: what you need to know, include, and add to your font's OpenType MATH table.
1. Core concept

Math layout engines (TeX, KaTeX, MathJax) build tall symbols by stacking three or more glyphs:

Top
Middle (extender)
Bottom

The font's MATH table defines which glyphs to use and how they connect.
2. Required glyphs

Each group must exist as real glyphs in the font (usually from Unicode block U+239B–U+23B3):


Group
Glyphs
Unicode
Names


Left parentheses
⎛ ⎜ ⎝
239B–239D
uni239B–uni239D


Right parentheses
⎞ ⎟ ⎠
239E–23A0
uni239E–uni23A0


Left square brackets
⎡ ⎢ ⎣
23A1–23A3
uni23A1–uni23A3


Right square brackets
⎤ ⎥ ⎦
23A4–23A6
uni23A4–uni23A6


Left curly braces
⎧ ⎨ ⎩
23A7–23A9
uni23A7–uni23A9


Right curly braces
⎫ ⎬ ⎭
23AB–23AD
uni23AB–uni23AD


Integral
⌠ ⌡ ⎲ ⎳
2320–23B3
uni2320–uni23B3


Summation top/bottom
⎰ ⎱
23B0–23B1
uni23B0–uni23B1


Bars / pipes
⎪ ⎮
23AA, 23D0
uni23AA, uni23D0


Ceiling / floor
⌈ ⌉ ⌊ ⌋
2308–230B
uni2308–uni230B


Each should share stroke weight, side bearings, and vertical join points.
3. Minimal MATH table (TTX format)

<MATH>
  <MathVariants>

    <!-- Parentheses -->
    <VertGlyphAssembly glyph="parenleft">
      <Part glyph="uni239B" fullAdvance="350"/>
      <Part glyph="uni239C" fullAdvance="400" isExtender="1"/>
      <Part glyph="uni239D" fullAdvance="350"/>
    </VertGlyphAssembly>
    <VertGlyphAssembly glyph="parenright">
      <Part glyph="uni239E" fullAdvance="350"/>
      <Part glyph="uni239F" fullAdvance="400" isExtender="1"/>
      <Part glyph="uni23A0" fullAdvance="350"/>
    </VertGlyphAssembly>

    <!-- Square brackets -->
    <VertGlyphAssembly glyph="bracketleft">
      <Part glyph="uni23A1" fullAdvance="350"/>
      <Part glyph="uni23A2" fullAdvance="400" isExtender="1"/>
      <Part glyph="uni23A3" fullAdvance="350"/>
    </VertGlyphAssembly>
    <VertGlyphAssembly glyph="bracketright">
      <Part glyph="uni23A4" fullAdvance="350"/>
      <Part glyph="uni23A5" fullAdvance="400" isExtender="1"/>
      <Part glyph="uni23A6" fullAdvance="350"/>
    </VertGlyphAssembly>

    <!-- Curly braces -->
    <VertGlyphAssembly glyph="braceleft">
      <Part glyph="uni23A7" fullAdvance="350"/>
      <Part glyph="uni23A8" fullAdvance="400" isExtender="1"/>
      <Part glyph="uni23A9" fullAdvance="350"/>
    </VertGlyphAssembly>
    <VertGlyphAssembly glyph="braceright">
      <Part glyph="uni23AB" fullAdvance="350"/>
      <Part glyph="uni23AC" fullAdvance="400" isExtender="1"/>
      <Part glyph="uni23AD" fullAdvance="350"/>
    </VertGlyphAssembly>

    <!-- Integrals -->
    <VertGlyphAssembly glyph="integral">
      <Part glyph="uni2320" fullAdvance="400"/>
      <Part glyph="uni23AE" fullAdvance="400" isExtender="1"/>
      <Part glyph="uni2321" fullAdvance="400"/>
    </VertGlyphAssembly>

    <!-- Summation (⎰⎱) -->
    <VertGlyphAssembly glyph="summationdisplay">
      <Part glyph="uni23B0" fullAdvance="400"/>
      <Part glyph="uni23B1" fullAdvance="400"/>
    </VertGlyphAssembly>

    <!-- Vertical bars -->
    <VertGlyphAssembly glyph="bar">
      <Part glyph="uni23AA" fullAdvance="400" isExtender="1"/>
    </VertGlyphAssembly>
    <VertGlyphAssembly glyph="doublebar">
      <Part glyph="uni23D0" fullAdvance="400" isExtender="1"/>
    </VertGlyphAssembly>

  </MathVariants>

  <MathConstants>
    <MathValue name="AxisHeight" value="0"/>
    <MathValue name="RadicalRuleThickness" value="40"/>
    <MathValue name="DisplayOperatorMinHeight" value="1200"/>
    <MathValue name="DelimiterPercent" value="90"/>
    <MathValue name="ScriptPercentScaleDown" value="80"/>
    <MathValue name="ScriptScriptPercentScaleDown" value="60"/>
    <MathValue name="MathLeading" value="100"/>
  </MathConstants>
</MATH>
4. Steps to use it


Decompile font
ttx MyFont.otf


Insert the XML snippet above (inside <MATH>).


Rebuild
ttx MyFont.ttx


Test
<style>.katex { font-family: "MyFont"; }</style>
<script>
  katex.render("\\left(\\begin{matrix}a\\\\b\\\\c\\\\d\\\\e\\\\f\\\\g\\\\h\\end{matrix}\\right)", el, {displayMode:true})
</script>


5. Design & metric notes


Vertical joins: each piece's top and bottom edges must align exactly.
Extender height ≈ 1 em; enough to repeat without visual break.
Side bearings: balanced (≈ 1/4 em for curved shapes).
No vertical padding beyond the actual glyph outline.
Stroke weight and curvature consistent across pieces.
Integrals and sums may be stylistic — just ensure top/bottom join neatly.

6. Verify visually

Use KaTeX or MathJax:
katex.render("\\left\\{ \\begin{matrix}a\\\\b\\\\c\\\\d\\\\e\\\\f\\\\g\\\\h\\end{matrix}\\right\\}", el)
All stacked delimiters should connect seamlessly, with balanced spacing and consistent thickness.
✅ Summary


Provide top, middle, bottom glyphs for each delimiter, define them in MATH > MathVariants, ensure precise bounding boxes, matching strokes, and balanced spacing.
This lets KaTeX/TeX dynamically assemble perfectly smooth, tall symbols for every case.

Monospaced Math Glyph Design — Making Tall Symbols Work Perfectly

When every glyph must occupy exactly one cell width, tall math delimiters can still look correct — but you must treat proportions, spacing, and joins with surgical precision. Here’s the complete, compact guide.
1. Width principle

All glyphs, including tall delimiters, must share a fixed advance width (e.g. 600 units).
However, the drawn shape should optically center within that width:

Place vertical strokes (|, [, ]) centered exactly.
Place curved glyphs (parentheses) slightly inward (≈ 5–8%) to appear balanced in monospaced grids.

2. Side bearings

Keep identical side bearings across related pairs:


Glyphs
Left / Right bearing


⎛ ⎝
equal left / right


⎞ ⎠
mirror of left parens


⎡ ⎣
equal


⎤ ⎦
equal


Use narrow visible shapes inside the full cell, not narrower advances.
3. Vertical alignment


Align all top/middle/bottom parts so vertical connectors meet exactly on the baseline grid.
The vertical advance (fullAdvance in the MATH table) must equal the actual join distance, not the entire em box.
Test stacking: ⎛⎜⎝, ⎡⎢⎣, ⎧⎨⎩.

4. Optical weight

Because monospaced text compresses context visually:

Thicken curved outlines slightly (≈ +5%) to match the apparent weight of vertical bars.
Ensure top and bottom strokes of ⎛⎝ curve inward, not outward — avoids “leaning”.

5. Consistent metrics


All vertical pieces (⎜, ⎢, ⎨, ⎮) share identical horizontal position and width inside the cell.
Use the same centerline x-position across all left/right delimiter sets.
Verify that stacked composites align pixel-perfect in raster and vector views.

6. MATH table tuning

Inside VertGlyphAssembly, keep:
<Part glyph="..." fullAdvance="X"/>
where X = exact vertical connection step, typically equal across all monospaced pieces (e.g. 400 units).
No negative connector lengths — rely on perfect box-to-box fit.
7. Testing checklist

Use KaTeX display mode with your monospaced font:
katex.render("\\left(\\begin{matrix}a\\\\b\\\\c\\\\d\\\\e\\\\f\\\\g\\\\h\\end{matrix}\\right)", el, {displayMode:true})
Confirm:

No visible gaps between segments.
Left/right pairs visually centered over text columns.
Multiple lines of parentheses stack evenly within the grid.

8. Optional refinement

To preserve full monospacing and elegant proportion:

Keep true monospace advance, but use slightly asymmetric contours so (x) and [x] look balanced.
Apply hinting zones or vertical metrics alignment so joins remain exact across renderers.

Summary:

Keep every glyph the same width, center or offset outlines for optical balance, ensure vertical joins meet precisely, unify metrics across pairs, and fine-tune curvature and stroke weight.
Done right, tall delimiters will look clean, aligned, and perfectly proportional — even in a fully monospaced grid.

Math Font Outline

Below is a clean, structured outline of the exact components required to make CrowMarkMono into a proper OpenType Math font.
This is the “math-font spec” broken into actionable work units.
No fluff, just the essentials.
1. Generate the full OpenType MATH table

This table contains three major blocks.
1.1 Math Constants

Hundreds of numeric parameters, including:

ScriptPercentScaleDown
ScriptScriptPercentScaleDown
RadicalVerticalGap
RadicalRuleThickness
FractionNumeratorShiftUp
FractionDenominatorShiftDown
FractionRuleThickness
UpperLimitGapMin
LowerLimitGapMin
SubscriptShiftDown
SuperscriptShiftUp
AccentBaseHeight
StretchStackTopShiftUp
IntegralTopShiftUp / BottomShiftDown
LargeOpVerticalGap
UnderbarVerticalGap
OverbarVerticalGap

(These values make MathML layout work.)
1.2 Glyph Assembly Rules

Required for stretchable glyphs (e.g., parentheses, brackets):

Top glyph
Bottom glyph
Middle extender
Repeatable extender
Alignment anchors

1.3 Glyph Variants

Multiple size variants of:

Parentheses
Brackets
Braces
Vertical bars
Integral signs
Summation / product symbols
Arrows / fences
Radical signs
Large operators
Script and calligraphic letters

MathML scales glyphs by switching between these variants.

2. Add all math symbols from U+2200–U+22FF

This is the full “Mathematical Operators” block.
2.1 Logic operators

∀ ∃ ∄ ∧ ∨ ⊤ ⊥ ¬
2.2 Set relations

∈ ∉ ∊ ∋ ∌
⊂ ⊃ ⊄ ⊅ ⊆ ⊇ ⊈ ⊉
∪ ∩ ∅
2.3 Order and comparison

≤ ≥ ≠ ≡ ≜ ≝
≲ ≳ ≪ ≫
≈ ∼ ∝ ∞
2.4 Arithmetic operators

± ∓ × ÷ ⋅ ∘ ∗
∑ ∏ ∐ ∫ ∮ ∯ ∰
2.5 Arrow operators

← ↑ → ↓ ↔ ↕ ↖ ↗ ↘ ↙
⇐ ⇑ ⇒ ⇓ ⇔
⇦ ⇧ ⇨ ⇩
and dozens more (full range)
2.6 Miscellaneous

⋮ ⋯ ⋰ ⋱
‰ ‱
⊕ ⊗ ⊘ ⊖
⊢ ⊣ ⊨
⋃ ⋂ ⋁ ⋀
A real math font needs all of them.

3. Add all Mathematical Alphanumeric Symbols

These are from U+1D400–1D7FF.
3.1 Latin


Bold
Italic
Bold Italic
Script
Bold Script
Fraktur
Bold Fraktur
Double-struck
Sans-serif
Sans-serif Bold
Sans-serif Italic
Sans-serif Bold Italic
Monospace

3.2 Greek


Bold
Italic
Bold Italic
Sans-serif Bold
Sans-serif Bold Italic

(There is no script Greek.)
3.3 Digits

Digits exist for:

Double-struck
Sans-serif
Sans-serif Bold
Monospace

MathML expects these alphabets for variables, constants, vector notation, and stylistic math.

4. Add Stretchy Operator Constructions

MathML requires stretchable versions of:
4.1 Fences / brackets


( )
[ ]
{ }
|
‖
⟨ ⟩
⟪ ⟫

4.2 Arrows

All directions including long, double, harpoons.
4.3 Integrals

∫ extended to any height.
4.4 Radicals

√ must stretch horizontally with extenders.
Requires:

radical base
radical bar
optional degree position anchor

4.5 Vertical operators

|
∥
fences for cases & matrices
4.6 Overbars / underbars

—
extenders for long accents
Each stretchy construction requires “parts” and rules defined in the MATH table.

5. Add Large Operator Variants

MathML expects multiple sizes of:
5.1 Summation / product symbols


∑
∏
∐
⋁ ⋂ ⋃

5.2 Integrals


∫ ∮ ∯
Double/triple integrals must scale
Slanted integrals
Loop integrals

5.3 Large brackets

Parentheses, braces, brackets in sizes 1–5 (minimum), plus extender mode.
5.4 Stretching comparison operators

Large versions of ≤ ≥ = ≠ etc. for display math.
These must have variant glyphs named and registered.

6. Define math constants and layout metrics

These numeric values are essential so MathML knows:

where superscripts go
where subscripts go
how big radicals are
thickness of fraction bars
spacing above/below operators
shifting of scripts
proper assembly of tall operators

This is almost always tuned manually.

7. Add diacritic anchors for math accents

Math accents differ from text accents.
7.1 Accents to support


˙ (dot)
¨ (double dot)
^ (hat)
ˇ (check)
˜ (tilde)
ˉ (overbar)
stretched ^
stretched ~
vector arrows above characters

7.2 Required anchor points

Every base glyph needs at least:

top accent anchor
bottom accent anchor
math top accent position
scriptstyle accent position

7.3 For wide accents

Anchors needed for assembly pieces:

left piece
extender piece
right piece

MathML builds long accents over long expressions using these anchors.

Final Deliverable Summary

Here is the full outline you asked for, reduced to its core:


Generate full OpenType MATH table
Math constants, glyph assemblies, variant sizes.


Add all math symbols U+2200–U+22FF
Logic, arithmetic, sets, arrows, operators.


Add all Mathematical Alphanumeric Symbols
Italic, bold, bold-italic, script, double-struck, Fraktur, sans-serif, monospace.


Implement stretchy operator constructions
Brackets, braces, bars, arrows, radicals, integrals.


Provide large operator variants
Summation, product, integrals, oversize brackets.


Define math metrics
Superscript/subscript offsets, fraction rules, radical thickness, spacing constants.


Add diacritic anchors
For math-specific accents and wide accents.


Math Font Construction

1. MATH TABLE CONSTRUCTION

The MATH table is mandatory. It has three major substructures.
1.1 Math Constants (must implement ALL)

You must define numeric values for:

ScriptPercentScaleDown
ScriptScriptPercentScaleDown
DelimitedSubFormulaMinHeight
DisplayOperatorMinHeight
RadicalVerticalGap
RadicalDisplayStyleVerticalGap
RadicalRuleThickness
RadicalExtraAscender
RadicalKernBeforeDegree
RadicalKernAfterDegree
RadicalDegreeBottomRaisePercent
SubscriptShiftDown
SubscriptTopMax
SubscriptBaselineDropMin
SuperscriptShiftUp
SuperscriptShiftUpCramped
SuperscriptBottomMin
SuperscriptBaselineDropMax
FractionNumeratorShiftUp
FractionNumeratorDisplayStyleShiftUp
FractionDenominatorShiftDown
FractionDenominatorDisplayStyleShiftDown
FractionNumeratorGapMin
FractionNumeratorDisplayStyleGapMin
FractionDenominatorGapMin
FractionDenominatorDisplayStyleGapMin
FractionRuleThickness
UnderbarVerticalGap
UnderbarRuleThickness
UnderbarExtraDescender
OverbarVerticalGap
OverbarRuleThickness
OverbarExtraAscender
RadicalDegreeBottomHeight
StackTopShiftUp
StackTopDisplayStyleShiftUp
StackBottomShiftDown
StackBottomDisplayStyleShiftDown
StackGapMin
StackDisplayStyleGapMin
StretchStackTopShiftUp
StretchStackBottomShiftDown
StretchStackGapAboveMin
StretchStackGapBelowMin
AxisHeight

These define how every math expression is built and spaced.
1.2 Glyph Variants (required)

For each glyph that may need to grow:

Parentheses: (, )
Brackets: [, ]
Braces: {, }
Vertical bars: |, ∥
Angle brackets: ⟨ ⟩ ⟪ ⟫
Integrals: ∫ ∮ ∯ ∱ ∲ ∳
Sum/product: ∑ ∏ ∐
Large operators: ⋃ ⋂ ⋁ ⋀
Arrows: → ← ↔ ⇒ ⇐ ⇔ and long forms
Radicals: √ (radical with extender)

For each, provide variant glyphs:

Size 1
Size 2
Size 3
Size 4
Size 5
Unlimited size via extender piece

And connect them with MATH table links.
1.3 Glyph Assembly (stretch instructions)

Each stretchable operator needs “parts”:

Top piece
Bottom piece
Middle piece
Extender (repeatable)

And anchors specifying:

Vertical alignment
Horizontal alignment
Extender repeat positions


2. MATH ALPHANUMERIC SYMBOLS (U+1D400–1D7FF)

2.1 Required Latin alphabets

You must include the following for A–Z and a–z:

Mathematical Bold
Mathematical Italic
Mathematical Bold Italic
Mathematical Script
Mathematical Bold Script
Mathematical Fraktur
Mathematical Bold Fraktur
Mathematical Double-Struck
Sans-serif
Sans-serif Bold
Sans-serif Italic
Sans-serif Bold Italic
Monospace

Total glyphs: ~1300+
2.2 Required Greek alphabets

These sets must be included:

Bold Greek
Italic Greek
Bold Italic Greek
Sans-serif Bold Greek
Sans-serif Bold Italic Greek

(Other Greek styles do not exist in Unicode.)
2.3 Required digits

Include digits in:

Double-struck
Sans-serif
Sans-serif Bold
Monospace


3. MATHEMATICAL OPERATORS (U+2200–U+22FF)

You must implement every glyph from the Unicode block:
3.1 Logic & quantifiers

∀ ∁ ∂ ∃ ∄ ∅ ∆ ∇
3.2 Set operators

∈ ∉ ∊ ∋ ∌
⊂ ⊃ ⊄ ⊅ ⊆ ⊇ ⊈ ⊉
∪ ∩ ∅
⋃ ⋂ ⋁ ⋀
3.3 Relations

= ≠ ≡ ≅ ≈ ≤ ≥ ≺ ≻ ≼ ≽
3.4 Operators

± ∓ × ÷ ⋅ ∘ ∗ ⊕ ⊖ ⊗ ⊘ ⊙
3.5 Arrows (ALL variants)

← ↑ → ↓ ↔
↦ ↤ ↦ ↣
⇐ ⇑ ⇒ ⇓ ⇔
↮ ↭
Long arrows: ⟶ ⟵ ⟷
Harpoons: ⇀ ⇁ ⇂ ⇃
Double arrows, triple arrows
3.6 Fractions

⅟ ½ ⅓ ¼ etc.
But more importantly:

Fraction slash (U+2044)
Division slash (U+2215)

3.7 Vertical ellipsis & diagonal ellipsis

⋮ ⋯ ⋰ ⋱
3.8 Miscellaneous operators

∞ ∝ ∠ ⦀ ⧴
⋒ ⋓ ⟂ ⟘ ⫯
(You must include all glyphs, even rarely used ones.)

4. STRETCHY OPERATOR CONSTRUCTIONS

These require assembly rules.
4.1 Vertical stretchy symbols


Parentheses
Brackets
Braces
Vertical bar
Double vertical bar
Fences (⟨ ⟩)
Cases brackets

4.2 Horizontal/stretchable symbols


Arrows (must stretch horizontally)
Long equal signs / equivalence signs
Overlines / underlines
Radical bar extension

4.3 Integral operators

Stretchable forms of:

∫
∬
∭
∮
⨏ (rare)

(Change slant angle and height.)

5. LARGE OPERATOR VARIANTS

Must include display-math sizes for:

∑
∏
∐
⋃
⋂
⋁
⋀
tall ∫ ∮ integrals
very tall parentheses
very tall brackets
oversized relation symbols (≤ ≥ = ≠)

These variants must be declared in the MATH table.

6. MATH METRICS & SPACING CONSTANTS

Every glyph involved in math must have:
6.1 Script positioning


top anchor (superscript)
bottom anchor (subscript)
cramped positioning values

6.2 Fraction metrics


numerator shift
denominator shift
rule thickness
min gaps

6.3 Radical metrics


radical rule thickness
extra ascender
degree kern before/after
percentage shift

6.4 Operator metrics


min height for display operator
vertical gaps around limits

6.5 Accent metrics


accent base height
superscript drop values

Without these, the browser falls back.

7. DIACRITIC ANCHORS FOR MATH ACCENTS

Math accents behave differently from text accents.
7.1 Base glyph anchors

For each glyph (a–z, Greek, etc.):

TopAnchor
BottomAnchor
TopAccentPosition
BottomAccentPosition

7.2 Accent glyph anchors

You must anchor:

hat (ˆ)
tilde (˜)
wide hat
wide tilde
macron
overline
dot
double dot
vector arrow
breve
caron

Each needs:

Left anchor
Right anchor
Extender anchor (for wide accents)

7.3 Wide accents (stretchy)

Provide extender pieces and:

left glyph
extender glyph
right glyph

MathML assembles these to stretch over long expressions.

Optional but Recommended

While not required, real math fonts also include:
A. Blackboard-bold Greek variants

B. Calligraphic alternatives

C. Larger-sized delimiters beyond size 5

D. Integral symbols with alternate slants

E. Double-integral and triple-integral stylistic sets
Group	Glyphs	Unicode	Names
Left parentheses	⎛ ⎜ ⎝	239B–239D	uni239B–uni239D
Right parentheses	⎞ ⎟ ⎠	239E–23A0	uni239E–uni23A0
Left square brackets	⎡ ⎢ ⎣	23A1–23A3	uni23A1–uni23A3
Right square brackets	⎤ ⎥ ⎦	23A4–23A6	uni23A4–uni23A6
Left curly braces	⎧ ⎨ ⎩	23A7–23A9	uni23A7–uni23A9
Right curly braces	⎫ ⎬ ⎭	23AB–23AD	uni23AB–uni23AD
Integral	⌠ ⌡ ⎲ ⎳	2320–23B3	uni2320–uni23B3
Summation top/bottom	⎰ ⎱	23B0–23B1	uni23B0–uni23B1
Bars / pipes	⎪ ⎮	23AA, 23D0	uni23AA, uni23D0
Ceiling / floor	⌈ ⌉ ⌊ ⌋	2308–230B	uni2308–uni230B
Glyphs	Left / Right bearing
`⎛` `⎝`	equal left / right
`⎞` `⎠`	mirror of left parens
`⎡` `⎣`	equal
`⎤` `⎦`	equal