Inter Math

Examples

Famous equations at the size they deserve. Each is set in Inter Math at the weight on its label, as large as the page allows.

No. 01 Inter Math Black · wght 900

\displaystyle \int_a^b f'(x)\, dx = f(b) - f(a)

The fundamental theorem of calculus

Isaac Newton and Gottfried Wilhelm Leibniz, independently, 1660s and 1670s

No. 02 Inter Math Thin · wght 100

\displaystyle E = mc^2

Mass–energy equivalence

Albert Einstein, 1905

No. 03 Inter Math Medium · wght 500

\displaystyle i\hbar\,\frac{\partial}{\partial t}\Psi(\vec{r}, t) = \hat{H}\,\Psi(\vec{r}, t)

The Schrödinger equation

Erwin Schrödinger, 1926

No. 04 Inter Math · wght 350

Gauss’s law

\displaystyle \htmlClass{chalk-yellow}{\nabla} \cdot \vec{E} = \frac{\rho}{\varepsilon_0}

Gauss’s law for magnetism

\displaystyle \htmlClass{chalk-yellow}{\nabla} \cdot \vec{B} = 0

Faraday’s law

\displaystyle \htmlClass{chalk-yellow}{\nabla} \times \vec{E} = -\frac{\partial \vec{B}}{\partial t}

Ampère’s law, with Maxwell’s term

\displaystyle \htmlClass{chalk-yellow}{\nabla} \times \vec{B} = \mu_0 \vec{J} + \mu_0 \varepsilon_0 \frac{\partial \vec{E}}{\partial t}

Maxwell’s equations

James Clerk Maxwell, 1865, in Oliver Heaviside’s vector form, 1884

No. 05 Inter Math ExtraLight · wght 200

\displaystyle \oint_{C} \left( L \, dx + M \, dy \right) = \iint_{D} \left( \frac{\partial M}{\partial x} - \frac{\partial L}{\partial y} \right) dx \, dy

Green’s theorem

George Green, 1828

No. 06wght 800

\displaystyle a^2 + b^2 = c^2

Pythagoras’ theorem

Known in Babylon a thousand years before Pythagoras (Plimpton 322, c. 1800 BC)

No. 07wght 300

\displaystyle \varphi = \frac{1 + \sqrt{5}}{2}

The golden ratio

Euclid’s extreme and mean ratio, Elements VI, c. 300 BC

No. 08wght 600

\displaystyle e^{i\pi} + 1 = 0

Euler’s identity

Leonhard Euler, after his Introductio in analysin infinitorum, 1748. Five constants, three operations, one equals sign

No. 09wght 400

\displaystyle \hat{f}(\xi) = \int_{-\infty}^{\infty} f(x)\, e^{-2\pi i x \xi}\, dx

The Fourier transform

Joseph Fourier, Théorie analytique de la chaleur, 1822

No. 10wght 200

\displaystyle S = k \log W

Boltzmann’s entropy

Ludwig Boltzmann, 1877, in Max Planck’s notation. Engraved on Boltzmann’s grave in Vienna

No. 11wght 500

\displaystyle n! \sim \sqrt{2\pi n}\left(\frac{n}{e}\right)^{n}

Stirling’s formula

Abraham de Moivre and James Stirling, 1730

No. 12wght 400

\displaystyle P(A \mid B) = \frac{P(B \mid A)\, P(A)}{P(B)}

Bayes’ theorem

Thomas Bayes, published 1763; Pierre-Simon Laplace, 1774

No. 13wght 700

\displaystyle \sigma_x \sigma_p \geq \frac{\hbar}{2}

The uncertainty principle

Werner Heisenberg, 1927; in this form Earle Kennard, 1927

No. 14wght 300

\displaystyle G_{\mu\nu} + \Lambda g_{\mu\nu} = \frac{8\pi G}{c^4}\, T_{\mu\nu}

Einstein’s field equations

Albert Einstein, 1915. Λ added in 1917, dropped in 1931

No. 15wght 600

\displaystyle (i\gamma^{\mu}\partial_{\mu} - m)\,\psi = 0

The Dirac equation

Paul Dirac, 1928. Westminster Abbey carries it as iγ·∂ψ = mψ

No. 16wght 400

\displaystyle f(a) = \frac{1}{2\pi i} \oint_{\gamma} \frac{f(z)}{z - a}\, dz

Cauchy’s integral formula

Augustin-Louis Cauchy, 1831

No. 17wght 400

\displaystyle \begin{CD} G @>{\varphi}>> H \\ @V{\pi}VV @AA{\iota}A \\ G / \ker \varphi @>{\sim}>> \operatorname{im} \varphi \end{CD}

The first isomorphism theorem

Every homomorphism is a quotient, then an isomorphism, then an inclusion: the square commutes

No. 18wght 500

\displaystyle \rho\left(\frac{\partial \vec{u}}{\partial t} + (\vec{u} \cdot \nabla)\,\vec{u}\right) = -\nabla p + \mu \nabla^{2} \vec{u} + \vec{f}

The Navier–Stokes equations

Claude-Louis Navier, 1822; George Stokes, 1845

Posts

Math made to be scrolled past, and stopped at: quizzes, puzzles and stories, upright for stories and reels, wide for the feed. The account and its posts are made up.

e^{\pi}

Algebra

Quick quiz

(2x - 5)^2 = \cdots

A $4x^2 - 25$
B $4x^2 + 5x + 25$
C $2x^2 - 20x + 5$
D $4x^2 - 20x + 25$

@intermath

e^{\pi}

Inter Math @intermath · 2h

Can you find $c$ in 15 seconds?

C

A

B

6

8

c

e^{\pi}

Inter Math @intermath · Oct 15

Most people get this wrong. Can you find the rule?

\begin{aligned} 4 + 2 &= 12 \\ 3 + 4 &= 14 \\ 5 + 3 &= 16 \\ 7 + 9 &= \,? \end{aligned}

Find the derivative of

Calculus

y = e^{x}

A $\dfrac{dy}{dx} = e^{x}$
B $\dfrac{dy}{dx} = x e^{x}$
C $\dfrac{dy}{dx} = e^{2x}$
D $\dfrac{dy}{dx} = 1$

Answer in the comments

The story goes

Gauss, aged 9, adds 1 to 100 in seconds.

1 + 2 + \cdots + 100 = 5050

\displaystyle \sum_{k=1}^{n} k = \frac{n(n+1)}{2}

@intermath

e^{\pi}

Inter Math @intermath · 5h

No calculator. Which is bigger?

e^{\pi} \;\; \htmlClass{chalk-yellow}{\gtrless} \;\; \pi^{e}

Papers

Math at reading size, in running text: a journal page, Einstein and Fourier retold, a lecture, a worksheet. Switch them to Computer Modern to compare: nothing moves but the glyphs.

Math in

The Specimen Journal of MathematicsVol. 1, No. 1 (2026), 1–2

On the sum of the reciprocals of the squares

Abstract We recall how Euler found $\zeta(2)$ , the sum of the reciprocals of the squares, from the roots of the sine, and state his value of $\zeta(2m)$ for every $m \geq 1$ .

1. The problem

Pietro Mengoli asked in 1650 for the value of

\sum_{n=1}^{\infty} \frac{1}{n^2} = 1 + \frac{1}{4} + \frac{1}{9} + \frac{1}{16} + \cdots \tag{1}

Jakob Bernoulli made it famous in 1689 and could not find it. It took the name of Basel, the town of the Bernoullis and of Euler.

2. Euler’s argument

The sine vanishes at $x = k\pi$ , $k \in \mathbb{Z}$ . Taking $\sin x / x$ for a polynomial of infinite degree with the roots $\pm k\pi$ , Euler wrote

\frac{\sin x}{x} = \prod_{k=1}^{\infty} \left( 1 - \frac{x^2}{k^2 \pi^2} \right). \tag{2}

Its series begins

\frac{\sin x}{x} = 1 - \frac{x^2}{3!} + \frac{x^4}{5!} - \cdots \tag{3}

and the coefficients of $x^2$ in (2) and (3) must agree:

\sum_{k=1}^{\infty} \frac{1}{k^2 \pi^2} = \frac{1}{6}, \quad\text{so}\quad \sum_{k=1}^{\infty} \frac{1}{k^2} = \frac{\pi^2}{6}. \tag{4}

Theorem 1 (Euler). For every integer $m \geq 1$ ,

\zeta(2m) = \sum_{n=1}^{\infty} \frac{1}{n^{2m}} = \frac{(-1)^{m+1} B_{2m} \, (2\pi)^{2m}}{2 \, (2m)!},

where $B_{2m}$ are the Bernoulli numbers: $B_2 = \tfrac{1}{6}$ , $B_4 = -\tfrac{1}{30}$ , and so on.

Remark. The product (2) waited more than a century for a proof: Weierstrass’s factorization theorem, 1876. For odd arguments no such formula is known; Apéry proved $\zeta(3)$ irrational only in 1978.

A readingAnnalen der Physik 18 (1905), 639–641 [10]

Does the inertia of a body depend on its energy content?

Albert Einstein’s argument of 1905, in his notation: $V$ is the speed of light, $L$ an energy.

A body at rest sends out two equal waves of light in opposite directions, at an angle $\varphi$ to the $x$ axis, of energy $L$ together. Its energy goes from $E_0$ to $E_1$ :

E_0 = E_1 + \tfrac{1}{2} L + \tfrac{1}{2} L.

Seen from a system moving along $x$ at the velocity $v$ , the body goes from $H_0$ to $H_1$ , and the light carries more:

\begin{aligned} H_0 &= H_1 + \tfrac{1}{2} L \, \frac{1 - \frac{v}{V} \cos\varphi}{\sqrt{1 - \left( \frac{v}{V} \right)^2}} + \tfrac{1}{2} L \, \frac{1 + \frac{v}{V} \cos\varphi}{\sqrt{1 - \left( \frac{v}{V} \right)^2}} \\ &= H_1 + \frac{L}{\sqrt{1 - \left( \frac{v}{V} \right)^2}}. \end{aligned}

The differences $H - E$ are the body’s kinetic energies $K$ in the moving system, up to a constant, so that

K_0 - K_1 = L \left\{ \frac{1}{\sqrt{1 - \left( \frac{v}{V} \right)^2}} - 1 \right\} = \frac{1}{2} \, \frac{L}{V^2} \, v^2 + \cdots

The body moves on as before, with less kinetic energy, as if its mass had dropped by $L / V^2$ .

If a body gives off the energy $L$ as radiation, its mass decreases by $L / V^2$ .

Write $E$ for $L$ and $c$ for $V$ : $E = mc^2$ .

A readingParis, 1822 [11]

The analytical theory of heat

Joseph Fourier’s series, retold.

Heat spreads along a thin rod as

\frac{\partial u}{\partial t} = \kappa \, \frac{\partial^2 u}{\partial x^2}.

Fourier’s idea: write the first temperatures $f$ as a sum of sines, for each of them only fades. With the ends kept at $0$ ,

u(x, t) = \sum_{n=1}^{\infty} b_n \, e^{-\kappa n^2 t} \sin nx,

b_n = \frac{2}{\pi} \int_0^{\pi} f(x) \sin nx \, dx.

Even a rod cold at its ends and hot all through is such a sum:

\frac{\pi}{4} = \sin x + \frac{\sin 3x}{3} + \frac{\sin 5x}{5} + \cdots \qquad (0 < x < \pi).

In the middle, at $x = \tfrac{\pi}{2}$ , it is Leibniz’s series:

\frac{\pi}{4} = 1 - \frac{1}{3} + \frac{1}{5} - \frac{1}{7} + \cdots

Lagrange doubted that a function with a jump could be a sum of sines. Dirichlet proved it, in 1829.

Lecture 3 · AnalysisThe Gaussian integral

\begin{aligned} I &= \int_{-\infty}^{\infty} e^{-x^2} \, dx \\[0.5em] I^2 &= \int_{-\infty}^{\infty} \! \int_{-\infty}^{\infty} e^{-(x^2 + y^2)} \, dx \, dy \\[0.5em] &= \int_{0}^{2\pi} \! \int_{0}^{\infty} e^{-r^2} \, \htmlClass{chalk-yellow}{r \, dr \, d\theta} \\[0.5em] &= 2\pi \left[ -\tfrac{1}{2} e^{-r^2} \right]_{0}^{\infty} = \pi \end{aligned}

\boxed{\; \int_{-\infty}^{\infty} e^{-x^2} \, dx = \sqrt{\pi} \;}

Analytic geometryWorksheet 4 · Exercise 2

In an orthonormal frame $(O;\, \vec{\imath},\, \vec{\jmath})$ , take the points $A(1;\,2)$ , $B(5;\,4)$ and $C(3;\,-2)$ .

1. Give the coordinates of $\overrightarrow{AB}$ and $\overrightarrow{AC}$ .

$\overrightarrow{AB} \begin{pmatrix} 4 \\ 2 \end{pmatrix}, \quad \overrightarrow{AC} \begin{pmatrix} 2 \\ -4 \end{pmatrix}$
2. Show that $\overrightarrow{AB} \cdot \overrightarrow{AC} = 0$ . What is the angle $\widehat{BAC}$ ?

$\overrightarrow{AB} \cdot \overrightarrow{AC} = 4 \times 2 + 2 \times (-4) = 0$ , so $\widehat{BAC} = 90^\circ$ .
3. Compute $AB$ and $AC$ . What is the triangle $ABC$ ?

$AB = \sqrt{4^2 + 2^2} = 2\sqrt{5} = AC$ : right-angled at $A$ , and isosceles.
4. $I$ is the midpoint of $[BC]$ . Give an equation of the circle through $A$ , $B$ and $C$ .

$I(4;\,1)$ and $IA = \sqrt{10}$ , so $(x - 4)^2 + (y - 1)^2 = 10$ .

Specimens

The workbench’s specimens, from the alphabet to the Navier–Stokes equations. In the list, set the weight and the size, lay Computer Modern alongside, or mark what was drawn for Inter Math.

Nine weights

Thin to Black, and every weight between: fraction bars, surds and arrows follow the weight, not only the glyphs.

100 Thin

\displaystyle \hat{f}(\xi) = \int_{-\infty}^{\infty} f(x)\, e^{-2\pi i x \xi}\, dx

200 ExtraLight

\displaystyle \hat{f}(\xi) = \int_{-\infty}^{\infty} f(x)\, e^{-2\pi i x \xi}\, dx

300 Light

\displaystyle \hat{f}(\xi) = \int_{-\infty}^{\infty} f(x)\, e^{-2\pi i x \xi}\, dx

400 Regular

\displaystyle \hat{f}(\xi) = \int_{-\infty}^{\infty} f(x)\, e^{-2\pi i x \xi}\, dx

500 Medium

\displaystyle \hat{f}(\xi) = \int_{-\infty}^{\infty} f(x)\, e^{-2\pi i x \xi}\, dx

600 SemiBold

\displaystyle \hat{f}(\xi) = \int_{-\infty}^{\infty} f(x)\, e^{-2\pi i x \xi}\, dx

700 Bold

\displaystyle \hat{f}(\xi) = \int_{-\infty}^{\infty} f(x)\, e^{-2\pi i x \xi}\, dx

800 ExtraBold

\displaystyle \hat{f}(\xi) = \int_{-\infty}^{\infty} f(x)\, e^{-2\pi i x \xi}\, dx

900 Black

\displaystyle \hat{f}(\xi) = \int_{-\infty}^{\infty} f(x)\, e^{-2\pi i x \xi}\, dx

Sizes

The browser picks the optical size from the font size: 14 for text, wider and looser; 32 for display, tighter and finer.

\{ x \in \mathbb{R} \}

opsz 14

\{ x \in \mathbb{R} \}

opsz 32

12 px

\displaystyle \sqrt{x^2 + y^2} \le \frac{|x| + |y|}{\sqrt{2}} \cdot 2

14 px

\displaystyle \sqrt{x^2 + y^2} \le \frac{|x| + |y|}{\sqrt{2}} \cdot 2

16 px

\displaystyle \sqrt{x^2 + y^2} \le \frac{|x| + |y|}{\sqrt{2}} \cdot 2

20 px

\displaystyle \sqrt{x^2 + y^2} \le \frac{|x| + |y|}{\sqrt{2}} \cdot 2

24 px

\displaystyle \sqrt{x^2 + y^2} \le \frac{|x| + |y|}{\sqrt{2}} \cdot 2

32 px

\displaystyle \sqrt{x^2 + y^2} \le \frac{|x| + |y|}{\sqrt{2}} \cdot 2

48 px

\displaystyle \sqrt{x^2 + y^2} \le \frac{|x| + |y|}{\sqrt{2}} \cdot 2

64 px

\displaystyle \sqrt{x^2 + y^2} \le \frac{|x| + |y|}{\sqrt{2}} \cdot 2

96 px

\displaystyle \sqrt{x^2 + y^2} \le \frac{|x| + |y|}{\sqrt{2}} \cdot 2

Weight400 Regular Size30 px

Letters and figures

Italic letters

abcdefghijklmnopqrstuvwxyz

\displaystyle abcdefghijklmnopqrstuvwxyz

Capitals

ABCDEFGHIJKLMNOPQRSTUVWXYZ

\displaystyle ABCDEFGHIJKLMNOPQRSTUVWXYZ

Figures

0123456789

\displaystyle 0123456789

Greek

\alpha\beta\gamma\delta\epsilon\varepsilon\zeta\eta\theta\vartheta\iota\kappa\lambda\mu\nu\xi\pi\varpi\rho\varrho\sigma\varsigma\tau\upsilon\phi\varphi\chi\psi\omega

\displaystyle \alpha\beta\gamma\delta\epsilon\varepsilon\zeta\eta\theta\vartheta\iota\kappa\lambda\mu\nu\xi\pi\varpi\rho\varrho\sigma\varsigma\tau\upsilon\phi\varphi\chi\psi\omega

Greek capitals

\Gamma\Delta\Theta\Lambda\Xi\Pi\Sigma\Upsilon\Phi\Psi\Omega

\displaystyle \Gamma\Delta\Theta\Lambda\Xi\Pi\Sigma\Upsilon\Phi\Psi\Omega

Look-alikes

l\,I\,1\,|\quad \mathrm{l\,I\,1}\quad O\,0\,o

\displaystyle l\,I\,1\,|\quad \mathrm{l\,I\,1}\quad O\,0\,o

Upright, bold, sans serif

\mathrm{d}x \quad \mathbf{u} + \boldsymbol{\alpha} \quad \mathsf{ABC\,xyz\,123}

\displaystyle \mathrm{d}x \quad \mathbf{u} + \boldsymbol{\alpha} \quad \mathsf{ABC\,xyz\,123}

Blackboard bold

\mathbb{N} \subset \mathbb{Z} \subset \mathbb{Q} \subset \mathbb{R} \subset \mathbb{C} \quad \mathbb{ABCDEFGHIJKLM}

\displaystyle \mathbb{N} \subset \mathbb{Z} \subset \mathbb{Q} \subset \mathbb{R} \subset \mathbb{C} \quad \mathbb{ABCDEFGHIJKLM}

Operators and relations

Binary operators

a + b - c \times d \div e \pm f \mp g \cdot h \ast i \star j \circ k \bullet l

\displaystyle a + b - c \times d \div e \pm f \mp g \cdot h \ast i \star j \circ k \bullet l

Relations

a = b \ne c \equiv d \approx e \cong f \sim g \simeq h \propto i \doteq j

\displaystyle a = b \ne c \equiv d \approx e \cong f \sim g \simeq h \propto i \doteq j

Order

a < b \le c \leqslant d \ll e \quad f > g \ge h \geqslant i \gg j \quad x \lesssim y \gtrapprox z

\displaystyle a < b \le c \leqslant d \ll e \quad f > g \ge h \geqslant i \gg j \quad x \lesssim y \gtrapprox z

Precedence, negations

a \prec b \preceq c \succ d \quad a \nless b \nleq c \quad A \nsubseteq B \quad 3 \nmid 7 \quad (d) \nparallel (d')

\displaystyle a \prec b \preceq c \succ d \quad a \nless b \nleq c \quad A \nsubseteq B \quad 3 \nmid 7 \quad (d) \nparallel (d')

Circled and boxed

a \oplus b \ominus c \otimes d \oslash e \odot f \quad a \boxplus b \boxminus c \boxtimes d \boxdot e

\displaystyle a \oplus b \ominus c \otimes d \oslash e \odot f \quad a \boxplus b \boxminus c \boxtimes d \boxdot e

Sets and logic

Sets

\{x \in \mathbb{R} \mid x > 0\} \quad A \cup B \cap C \setminus D \quad A \subseteq B \subsetneq C \quad \emptyset \ne \varnothing

\displaystyle \{x \in \mathbb{R} \mid x > 0\} \quad A \cup B \cap C \setminus D \quad A \subseteq B \subsetneq C \quad \emptyset \ne \varnothing

Quantifiers

\forall \varepsilon > 0\ \exists \delta > 0 : |x - a| < \delta \Rightarrow |f(x) - f(a)| < \varepsilon

\displaystyle \forall \varepsilon > 0\ \exists \delta > 0 : |x - a| < \delta \Rightarrow |f(x) - f(a)| < \varepsilon

Logic

\Gamma \vdash \varphi \quad \Gamma \models \psi \quad p \land q \lor \lnot r \quad \top \; \bot \quad \therefore \; \because

\displaystyle \Gamma \vdash \varphi \quad \Gamma \models \psi \quad p \land q \lor \lnot r \quad \top \; \bot \quad \therefore \; \because

Arrows

\leftarrow \rightarrow \leftrightarrow \Leftarrow \Rightarrow \Leftrightarrow \mapsto \hookrightarrow \uparrow \downarrow \nearrow \searrow

\displaystyle \leftarrow \rightarrow \leftrightarrow \Leftarrow \Rightarrow \Leftrightarrow \mapsto \hookrightarrow \uparrow \downarrow \nearrow \searrow

Long and stretched

A \longrightarrow B \Longleftrightarrow C \quad A \xrightarrow{\ f\ } B \xleftarrow[\text{below}]{\text{above}} C

\displaystyle A \longrightarrow B \Longleftrightarrow C \quad A \xrightarrow{\ f\ } B \xleftarrow[\text{below}]{\text{above}} C

Harpoons, pairs, curves

\rightleftharpoons \leftrightarrows \rightrightarrows \twoheadrightarrow \rightarrowtail \rightsquigarrow \curvearrowright \circlearrowleft

\displaystyle \rightleftharpoons \leftrightarrows \rightrightarrows \twoheadrightarrow \rightarrowtail \rightsquigarrow \curvearrowright \circlearrowleft

Delimiters

Sizes

\Biggl( \biggl( \Bigl( \bigl( ( \; ) \bigr) \Bigr) \biggr) \Biggr) \quad \Biggl\{ \biggl[ \Bigl\langle \bigl| \; \bigr| \Bigr\rangle \biggr] \Biggr\}

\displaystyle \Biggl( \biggl( \Bigl( \bigl( ( \; ) \bigr) \Bigr) \biggr) \Biggr) \quad \Biggl\{ \biggl[ \Bigl\langle \bigl| \; \bigr| \Bigr\rangle \biggr] \Biggr\}

Around fractions

\left( \frac{1}{2} \right) \left[ \frac{\sqrt{2}}{2} \right] \left\{ \frac{a}{b} \right\} \left\langle \frac{u}{v} \right\rangle \left\lfloor \frac{x}{2} \right\rfloor \left\lceil \frac{y}{3} \right\rceil

\displaystyle \left( \frac{1}{2} \right) \left[ \frac{\sqrt{2}}{2} \right] \left\{ \frac{a}{b} \right\} \left\langle \frac{u}{v} \right\rangle \left\lfloor \frac{x}{2} \right\rfloor \left\lceil \frac{y}{3} \right\rceil

Bars and double brackets

|x| \quad \|v\| \quad \llbracket a, b \rrbracket \quad \lBrace x \rBrace \quad \left| \frac{a}{b} \right| \quad \left\| \frac{\vec{u}}{2} \right\|

\displaystyle |x| \quad \|v\| \quad \llbracket a, b \rrbracket \quad \lBrace x \rBrace \quad \left| \frac{a}{b} \right| \quad \left\| \frac{\vec{u}}{2} \right\|

Matrices

\begin{pmatrix} a & b \\ c & d \end{pmatrix} \begin{bmatrix} 1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 1 \end{bmatrix} \begin{vmatrix} x & y \\ z & w \end{vmatrix}

\displaystyle \begin{pmatrix} a & b \\ c & d \end{pmatrix} \begin{bmatrix} 1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 1 \end{bmatrix} \begin{vmatrix} x & y \\ z & w \end{vmatrix}

Cases

|x| = \begin{cases} x & \text{if } x \ge 0 \\ -x & \text{if } x < 0 \end{cases}

\displaystyle |x| = \begin{cases} x & \text{if } x \ge 0 \\ -x & \text{if } x < 0 \end{cases}

Big operators

Display

\sum_{k=1}^{n} k^2 \quad \prod_{p} \frac{1}{1 - p^{-s}} \quad \int_0^1 f(x)\,dx \quad \oint_C \omega \quad \iint_D \quad \iiint_V

\displaystyle \sum_{k=1}^{n} k^2 \quad \prod_{p} \frac{1}{1 - p^{-s}} \quad \int_0^1 f(x)\,dx \quad \oint_C \omega \quad \iint_D \quad \iiint_V

In text

\textstyle \sum_{k=1}^{n} k = \frac{n(n+1)}{2}, \quad \int_0^1 x^2\,dx = \frac{1}{3}, \quad \prod_{i} x_i

\displaystyle \textstyle \sum_{k=1}^{n} k = \frac{n(n+1)}{2}, \quad \int_0^1 x^2\,dx = \frac{1}{3}, \quad \prod_{i} x_i

Sets and logic

\bigcup_{i \in I} A_i \quad \bigcap_{n} B_n \quad \bigoplus_k V_k \quad \bigotimes_j W_j \quad \bigvee_i p_i \quad \bigwedge_j q_j

\displaystyle \bigcup_{i \in I} A_i \quad \bigcap_{n} B_n \quad \bigoplus_k V_k \quad \bigotimes_j W_j \quad \bigvee_i p_i \quad \bigwedge_j q_j

Fractions and roots

Fractions

\frac{1}{2} + \frac{x + 1}{x - 1} = \frac{\frac{a}{b}}{c} \quad \tfrac{3}{4} \quad \binom{n}{k}

\displaystyle \frac{1}{2} + \frac{x + 1}{x - 1} = \frac{\frac{a}{b}}{c} \quad \tfrac{3}{4} \quad \binom{n}{k}

Continued

\sqrt{2} = 1 + \cfrac{1}{2 + \cfrac{1}{2 + \cfrac{1}{2 + \cdots}}}

\displaystyle \sqrt{2} = 1 + \cfrac{1}{2 + \cfrac{1}{2 + \cfrac{1}{2 + \cdots}}}

Roots

\sqrt{2} \quad \sqrt{x^2 + y^2} \quad \sqrt[3]{8} = 2 \quad \sqrt{\frac{a}{b}} \quad \sqrt{1 + \sqrt{1 + \sqrt{1 + x}}}

\displaystyle \sqrt{2} \quad \sqrt{x^2 + y^2} \quad \sqrt[3]{8} = 2 \quad \sqrt{\frac{a}{b}} \quad \sqrt{1 + \sqrt{1 + \sqrt{1 + x}}}

Scripts and accents

Scripts

x^2 \quad x_i \quad x_i^2 \quad e^{-x^2} \quad 2^{2^{n}} \quad a_{n+1} = 2a_n - 1 \quad 10^{-3}

\displaystyle x^2 \quad x_i \quad x_i^2 \quad e^{-x^2} \quad 2^{2^{n}} \quad a_{n+1} = 2a_n - 1 \quad 10^{-3}

Primes and accents

f'(x),\ f''(x) \quad \hat{a}\ \tilde{n}\ \bar{x}\ \dot{y}\ \ddot{z}\ \vec{v}\ \check{c}\ \breve{u} \quad \mathring{A}

\displaystyle f'(x),\ f''(x) \quad \hat{a}\ \tilde{n}\ \bar{x}\ \dot{y}\ \ddot{z}\ \vec{v}\ \check{c}\ \breve{u} \quad \mathring{A}

Wide accents

\widehat{ABC} \quad \widetilde{xyz} \quad \overline{AB} \quad \underline{CD} \quad \overrightarrow{AB} \quad \overleftrightarrow{CD}

\displaystyle \widehat{ABC} \quad \widetilde{xyz} \quad \overline{AB} \quad \underline{CD} \quad \overrightarrow{AB} \quad \overleftrightarrow{CD}

Braces

\overbrace{a + b + \cdots + z}^{26} \quad \underbrace{1 + 1 + \cdots + 1}_{n\ \text{times}}

\displaystyle \overbrace{a + b + \cdots + z}^{26} \quad \underbrace{1 + 1 + \cdots + 1}_{n\ \text{times}}

In use

Geometry

\widehat{ABC} = 45^\circ \quad (AB) \parallel (CD) \quad (AB) \perp (EF) \quad \triangle ABC \sim \triangle DEF

\displaystyle \widehat{ABC} = 45^\circ \quad (AB) \parallel (CD) \quad (AB) \perp (EF) \quad \triangle ABC \sim \triangle DEF

Analysis

\lim_{x \to 0} \frac{\sin x}{x} = 1 \qquad f(x) = \sum_{n=0}^{\infty} \frac{f^{(n)}(a)}{n!}(x - a)^n

\displaystyle \lim_{x \to 0} \frac{\sin x}{x} = 1 \qquad f(x) = \sum_{n=0}^{\infty} \frac{f^{(n)}(a)}{n!}(x - a)^n

Text in math

\text{if } x \in \mathbb{Q} \text{, then } x^2 \in \mathbb{Q} \quad \text{area} = \pi r^2

\displaystyle \text{if } x \in \mathbb{Q} \text{, then } x^2 \in \mathbb{Q} \quad \text{area} = \pi r^2

Symbols

\infty\ \partial\ \nabla\ \hbar\ \ell\ \angle\ \triangle\ \square\ \blacksquare\ \checkmark\ \bigstar\ \spadesuit\heartsuit\diamondsuit\clubsuit\ \flat\natural\sharp

\displaystyle \infty\ \partial\ \nabla\ \hbar\ \ell\ \angle\ \triangle\ \square\ \blacksquare\ \checkmark\ \bigstar\ \spadesuit\heartsuit\diamondsuit\clubsuit\ \flat\natural\sharp

Glyphs

Every glyph says where it comes from. Pick one: the build’s own record tells what of Inter’s it is drawn from, and how.

Greek

Operators

Relations

Sets and logic

Arrows

Calculus

Blackboard bold

Delimiters

Shapes and signs

Still Computer Modern

Arrowheads

The small arrows come with two heads. Open angles by default, like every other arrow of the font; or filled triangles, like Inter’s own small arrows, with one option. Both are in the font: a stylistic set switches them.

Anglesdefault

\circlearrowleft \; \rightrightarrows \; \rightarrowtail \; \twoheadrightarrow \; \Lsh

\vec{F} = m\vec{a} \qquad \overrightarrow{AB} + \overrightarrow{BC} = \overrightarrow{AC}

setMathFont("inter")

Trianglesss01

\circlearrowleft \; \rightrightarrows \; \rightarrowtail \; \twoheadrightarrow \; \Lsh

\vec{F} = m\vec{a} \qquad \overrightarrow{AB} + \overrightarrow{BC} = \overrightarrow{AC}

setMathFont("inter", { arrowHeads: "triangle" })

Craft

Every math symbol is drawn from Inter: its strokes, curves and terminals, at every weight. What each symbol is, how large, and where it sits, comes from Computer Modern. So formulas look like Inter, and lay out like TeX.

Built from Inter’s parts

What Inter lacks is made of what it has: its letters turned, its bars and bowls taken apart and put together at the operators’ stroke. Each glyph records what it is drawn from, and the build graphs it.

\forall

\exists

\nabla

\cap

\in

\vdash

\models

\equiv

\emptyset

\subseteq

\notin

\leqslant

Sized by TeX, stroked by Inter

Inter draws its plus for text. A math plus is as wide as a capital is tall: Inter Math’s is half as large again, as TeX’s, but drawn at the weight that keeps Inter’s stroke. Scaled, it would be heavier.

Inter’s +

Scaled 1.5×

+

Inter Math’s

One axis, one gap

The operators sit on one axis, between the lowercase and the capitals, and fraction bars line up with the minus. The gap of the equals sign is the gap between every pair of parallel strokes: ≡ ≈ ∥ ⇒ ⊨.

-\frac{a}{b} + c = d \equiv e \Rightarrow f \times g

Scripts drawn for their size

KaTeX shrinks scripts to 70 and 50 percent, and shrunk glyphs get thin and tight. Inter Math gives them their own forms, after Inter’s fraction figures, through stylistic sets: copy $x^2$ and you still get x^2.

e^{-x^2} 2^{2^{n}}

Shrunk

e^{-x^2} 2^{2^{n}}

Drawn for the size

Blackboard bold after STIX Two

STIX Two Math’s double-struck construction on Inter’s capitals: the strokes a seriffed capital makes thick are doubled, each one a closed hollow with one gap, narrowing past Regular.

\mathbb{NZQRC}

100

\mathbb{NZQRC}

400

\mathbb{NZQRC}

900

Accents where Inter puts them

A math accent over an italic letter is Inter’s own accented italic letter. Inter Math in yellow, laid over Inter’s text in green: where they coincide, one colour, and no fringe.

\hat{a}

\tilde{n}

\ddot{u}

\acute{e}

Lines that follow the weight

A fraction bar is as thick as the minus; the surd’s bar as its strokes; over- and underlines, strike-throughs and cancels are Inter’s own. KaTeX’s images, the surds, wide accents, braces and stretched arrows, are redrawn: 58 of them.

\sqrt{\frac{a}{b}} \; \overline{AB} \; \cancel{5}

100

\sqrt{\frac{a}{b}} \; \overline{AB} \; \cancel{5}

900

Checked at every weight

Every sign’s stroke is measured against Inter’s own at every weight, and must come out at 1.00. Every change is looked at from 100 to 900 in one picture, laid over Inter’s text where math sets what text sets, at small and large sizes, at both optical sizes, on paper and on the board. KaTeX’s layout is tested in both fonts.

\forall \in \Rightarrow

100

\forall \in \Rightarrow

200

\forall \in \Rightarrow

300

\forall \in \Rightarrow

400

\forall \in \Rightarrow

500

\forall \in \Rightarrow

600

\forall \in \Rightarrow

700

\forall \in \Rightarrow

800

\forall \in \Rightarrow

900

What stays Computer Modern

Inter has no typewriter, Fraktur, calligraphic or script letters, and no ℜ ℑ ℘ ℵ ℶ ℷ ℸ. Inter Math doesn’t invent them: they stay in KaTeX’s own fonts, with KaTeX’s metrics.

\mathcal{L} \; \mathfrak{g} \; \mathscr{F} \; \mathtt{x} \; \aleph_0 \; \wp \; \Re

Use with KaTeX

KaTeX lays the formulas out, with the rules of TeX. Inter Math gives it its glyphs and their measures, and a way back to Computer Modern.

KaTeX sets TeX on the web, fast, to the rules Donald Knuth wrote for TeX. It places every glyph by its measures: how wide, how tall, how deep. Its own fonts are Computer Modern.

Inter Math takes their place: upright and italic letters, large delimiters and operators, blackboard bold, the AMS symbols. It comes with the measures KaTeX needs at every weight, and with a small patch to KaTeX for what a font can’t say: how thick a fraction bar is, how a root is drawn.

One call switches back: setMathFont("classic") restores KaTeX exactly as it was. Try it:

400

\displaystyle \left( \sum_{k=1}^{n} a_k b_k \right)^{2} \le \left( \sum_{k=1}^{n} a_k^{2} \right) \left( \sum_{k=1}^{n} b_k^{2} \right)

Weight

1pnpm add inter-math katex@0.19.0, and Inter Math’s KaTeX patch (pnpm’s patchedDependencies).
2KaTeX’s CSS, then Inter Math’s.
3The inter-math class on an ancestor of the math, and its weight.
4setMathFont("inter", { weight }) before rendering, at that weight.

// pnpm add inter-math katex@0.19.0
import "katex/dist/katex.min.css";
import "inter-math/inter-math.css";
import katex from "katex";
import { setMathFont } from "inter-math";

setMathFont("inter", { weight: 400 });
katex.render("e^{i\\pi} + 1 = 0", element);

// on an ancestor of element:
// class="inter-math"
// style="--inter-math-weight: 400"

References

The type, the software, and the works the examples come from.

Type and software

[1]Rasmus Andersson, Inter, typeface, version 4.001. rsms.me/inter. SIL Open Font License 1.1. Inter Math is drawn from it, independently of its author.
[2]KaTeX 0.19.0, Khan Academy and contributors. katex.org. MIT License.
[3]Donald E. Knuth, The TeXbook, Addison-Wesley, 1984. Appendix G: how formulas are laid out.
[4]Donald E. Knuth, Computer Modern Typefaces, Computers and Typesetting E, Addison-Wesley, 1986.
[5]STIX Two fonts. stixfonts.org. SIL Open Font License 1.1. The reference for the blackboard letters.
[6]AMSFonts, American Mathematical Society. The AMS symbols KaTeX sets.
[7]fontTools. github.com/fonttools/fonttools. MIT License. Inter Math’s build.

Works

[8]Leonhard Euler, Introductio in analysin infinitorum, vol. 1, Lausanne, 1748, §138.
[9]Leonhard Euler, De summis serierum reciprocarum, Commentarii academiae scientiarum Petropolitanae 7 (1740), 123–134 (E41).
[10]Albert Einstein, Ist die Trägheit eines Körpers von seinem Energieinhalt abhängig?, Annalen der Physik 18 (1905), 639–641.
[11]Joseph Fourier, Théorie analytique de la chaleur, Firmin Didot, Paris, 1822.
[12]James Clerk Maxwell, A Dynamical Theory of the Electromagnetic Field, Philosophical Transactions of the Royal Society of London 155 (1865), 459–512.
[13]Erwin Schrödinger, Quantisierung als Eigenwertproblem, Annalen der Physik 79 (1926), 361–376.
[14]Srinivasa Ramanujan, letter to G. H. Hardy, 16 January 1913. G. H. Hardy, Ramanujan: Twelve Lectures on Subjects Suggested by His Life and Work, Cambridge University Press, 1940, p. 9.