I'm a professor of mathematics at the University of North Carolina Asheville. I've also done a fair amount of consulting work over the years focusing on scientific and data visualization.
I'm happy to be part of the first Observable Ambassador's cohort.
Public
Edited
Mar 30, 2023
3 stars
A Julia set on the Riemann sphereThe Z-CurveBarnsley's fernA stochastic digraph IFS algorithmSelf-affine tilesThe TwindragonThe Eisenstein fractionsA self-affine tile with holesSelf-affine tiles via polygon mergeGolden rectangle fractalsBifurcation diagram with critical curvesThe tame twindragonIllustrations for the proof of Green's theoremNon-orientability of a Mobius stripExamples of parametric surfacesPenrose tilingThe extended unit circlePenrose three coloringNewtons's method on the Riemann sphereConic sectionsDivisor graphsThe dance of Earth and VenusIterating multiples of the sine functionBorderline fractalsSelf-similar intersectionsBox-counting dimension examplesMandelbrot by dimensionInverse iteration for quadratic Julia setsInteger Apollonian PackingsIllustrations of two-dimensonal heat flowThe logistic bifurcation locusThe eleven unfoldings of the cubeA unimodal function with fractal level curvesGreen's theorem and polygonal areaThe geometry and numerics of first order ODEsThe xxx^xxx-spindleAnimated beatsRauzy FractalsHilbert's coordinate functionsPluckNot PiDrum strikeThe Koch snowflakeFractalized squareA Taylor series about π/4\pi/4π/4PlotX3D HyperboloidA PlotX3D animationModular arithmetic in 5th grade artSimple S-I-R ModelThe Poisson KernelPoly-gasketsClassification of 2D linear systems via trace and determinantJulia sets and the Mandelbrot setWater wavesFourier SeriesDisks for a solid of revolutionOrbit detection for the Mandelbrot setTracing a path on a spherePlot for mathematiciansFunctions of two variablesPartial derivativesDijkstra's algorithm on an RGGGradient ascentUnfolding polyhedraTangent plane to a level surfaceA strange discontinuityExamples of level surfacesMcMullen carpetsHills and valleysDouble and iterated integralsMST in an RGGTrees are bipartiteFractal typesettingd3.hierarchy and d3.treeK23 is PlanarPolar CoordinatesParametric region generatorParametric Plot 2DContour plotsGreedy graph coloringGraph6A few hundred interesting graphsThe Kings ProblemFirst order, autonomous systems of ODEsRunge-Kutta for systems of ODEs
The definition of ⇒
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Teaching
truthTableFunction([1, 1, 1, 1])("anything->q")
trueTruthTable = truthTableFunction()
function truth_slider(value, label) {
let seqs = [[0], [1]];
for (let i = 0; i < 3; i++) {
seqs = seqs.map((a) => [a.concat([0]), a.concat([1])]).flat();
}
let bits = d3.create("div");
let bitElem = bits.node();
bitElem.value = value;
let cur_value = seqs.map((c) => test(c, value)).indexOf(true);
bits.append("div").style("font-weight", "bold").html(label);
let step = bits.append("button").text("^");
let slider = bits
.append("input")
.attr("type", "range")
.attr("min", 0)
.attr("max", 15)
.attr("step", 1)
.attr("value", cur_value)
.style("margin-left", "10px");
let bit_display = bits
.append("div")
.style("display", "inline-block")
.style("margin-left", "10px")
.html(`[${value.map((t) => (t ? "T" : "F"))}]`);
slider.on("input", function () {
cur_value = parseInt(this.value);
bitElem.value = seqs[cur_value];
bit_display.html(`[${seqs[cur_value].map((d) => (d == 1 ? "T" : "F"))}]`);
bitElem.dispatchEvent(new CustomEvent("input"));
});
step.on("click", function () {
cur_value = (parseInt(slider.node().value) + 1) % 16;
slider.node().value = cur_value;
bitElem.value = seqs[cur_value];
bit_display.html(`[${seqs[cur_value].map((d) => (d == 1 ? "T" : "F"))}]`);
bitElem.dispatchEvent(new CustomEvent("input"));
});
return bitElem;
function test(a, b) {
return a[0] == b[0] && a[1] == b[1] && a[2] == b[2] && a[3] == b[3];
}
}
<style>
th:first-child input, td:first-child input{
display: none;
}
th {
pointer-events: none;
}
</style>
function truthTableFunction(bits = [1, 0, 1, 1]) {
function truthTable(exp) {
const data = [];
const sub = [];
const f = logicParser(exp);
const text = textFormula(f);
const subText = {};
const subPush = (f) => {
if (subText.hasOwnProperty(f.text)) return;
sub.push(f);
subText[f.text] = true;
};
function trav(f) {
if (f.node === "Literal") {
subPush(f);
return;
}
if (f.node === "Not") {
trav(f.child[0]);
subPush(f);
return;
}
trav(f.child[0]);
trav(f.child[1]);
subPush(f);
}
trav(f);
const alignment = {};
sub.forEach((f) => (alignment[f.text] = "center"));
let vars = variables(f);
for (let v of values(vars)) {
const row = {};
for (let i = 0; i < sub.length; ++i) {
row[sub[i].text] = evalFormula(sub[i], v) ? "T" : "F";
}
data.push(row);
}
return Inputs.table(data, { layout: "auto", align: alignment });
}
function textFormula(f) {
const precedence = {
Or: 2,
And: 2,
Implies: 3,
WeakImply: 3,
Not: 1,
Literal: 0
};
const paren = (f, op) =>
precedence[f.node] >= precedence[op]
? "(" + textFormula(f) + ")"
: textFormula(f);
if (f.text === undefined) {
switch (f.node) {
case "Literal":
f.text = f.value;
break;
case "Or":
f.text = paren(f.child[0], "Or") + "∨" + paren(f.child[1], "Or");
break;
case "Xor":
f.text = paren(f.child[0], "Xor") + " ⊕" + paren(f.child[1], "Xor");
break;
case "And":
f.text = paren(f.child[0], "And") + "∧" + paren(f.child[1], "And");
break;
case "Implies":
f.text =
paren(f.child[0], "Implies") + "⇒" + paren(f.child[1], "Implies");
break;
case "Equiv":
f.text =
paren(f.child[0], "Equiv") + "⇔" + paren(f.child[1], "Equiv");
break;
case "Not":
f.text = "¬" + paren(f.child[0], "Not");
break;
case "WeakImply":
f.text =
paren(f.child[0], "WeakImply") +
"→" +
paren(f.child[1], "WeakImply");
break;
}
}
return f.text;
}
function* values(vars) {
const n = vars.length;
for (let i = 0; i < 2 ** n; ++i) {
let vl = {};
let b = 1;
for (let j = n - 1; j >= 0; --j, b *= 2) {
vl[vars[j]] = (b & i) == 0;
}
yield vl;
}
}
function variables(f) {
const v = {};
function vr(f) {
if (f.node === "Literal") {
v[f.value] = true;
return;
}
f.child.forEach((c) => vr(c));
}
vr(f, v);
return Object.keys(v).sort();
}
function evalFormula(f, v) {
if (typeof f === "string") f = logicParser(f);
function ef(f) {
if (f.node === "Literal") return v[f.value];
if (f.node === "And") return ef(f.child[0]) && ef(f.child[1]);
if (f.node === "Or") return ef(f.child[0]) || ef(f.child[1]);
if (f.node === "Implies") return !ef(f.child[0]) || ef(f.child[1]);
if (f.node === "Equiv") return ef(f.child[0]) === ef(f.child[1]);
if (f.node === "Xor") {
const a = ef(f.child[0]);
const b = ef(f.child[1]);
return (a || b) && !(a && b);
}
if (f.node === "Not") return !ef(f.child[0]);
if (f.node === "WeakImply") {
return explicit_formula(...bits)(ef(f.child[0]), ef(f.child[1]));
}
}
return ef(f);
}
const expParserData = {
tokens: {
Space: /\s+/,
Literal: /\w+/,
Or: [/∨/, /\|/],
And: [/∧/, /&/],
Implies: [/⇒/, /=>/],
WeakImply: [/→/, /->/],
Equiv: [/⇔/, /<=>/],
Xor: [/⊕/, /\^/],
Not: [/¬/, /\!/],
LeftParen: /\(/,
RightParen: /\)/
},
binaryOp: ["Or", "And", "Implies", "WeakImply", "Equiv", "Xor"],
precedence: {
Or: 3,
And: 3,
Xor: 3,
Implies: 2,
WeakImply: 2,
Not: 4,
Equiv: 1
},
associativity: { Implies: "Right", Equiv: "Right", WeakImply: "Right" },
unaryOp: ["Not"],
terminal: ["Literal"]
};
let logicParser = expressionParser(expParserData);
function tokenizer(lex) {
const lexa = Object.entries(lex);
lex = [];
lexa.forEach(([a, b]) => {
if (b instanceof RegExp)
lex.push([new RegExp("^(" + b.source + ")(.*)"), a]);
else
b.forEach((t) => lex.push([new RegExp("^(" + t.source + ")(.*)"), a]));
});
return (str) => {
let s = str;
const r = [];
while (s) {
let flag = true;
for (let i = 0; i < lex.length; ++i) {
let m = s.match(lex[i][0]);
if (m) {
const l = lex[i][1];
r.push([l, m[1]]);
s = m[2];
flag = false;
break;
}
}
if (flag) throw "SyntaxError: invalid token";
}
r.push(["End", ""]);
return r;
};
}
function expressionParser(data) {
const to = tokenizer(data.tokens);
const binaryOp = {};
data.binaryOp.forEach((op) => (binaryOp[op] = true));
const isBinary = (tok) => binaryOp.hasOwnProperty(tok);
const unaryOp = {};
data.unaryOp.forEach((op) => (unaryOp[op] = true));
const isUnary = (tok) => unaryOp.hasOwnProperty(tok);
const terminal = {};
data.terminal.forEach((op) => (terminal[op] = true));
const isTerminal = (tok) => terminal.hasOwnProperty(tok);
const prec = (tok) => data.precedence[tok];
const associativity = (tok) =>
!data.associativity.hasOwnProperty(tok)
? "Left"
: data.associativity[tok];
return function (str) {
const l = to(str).filter(([tok, _]) => tok !== "Space");
let it = 0;
const next = () => l[it][0];
const value = () => l[it][1];
const consume = () => (it = it + 1);
const expect = (tok) => {
if (next() === tok) consume();
else throw "SyntaxError: unexpected token";
};
let Eparser, Exp, P;
const binary = (tok) => {
return tok;
};
const unary = (tok) => {
if (isBinary(tok)) return "Unary" + tok;
else return tok;
};
Eparser = () => {
const t = Exp(0);
expect("End");
return t;
};
Exp = (p) => {
let t = P();
while (isBinary(next()) && prec(binary(next())) >= p) {
const op = binary(next());
consume();
const q = prec(op) + (associativity(op) === "Right" ? 0 : 1);
const t1 = Exp(q);
t = { node: op, child: [t, t1] };
}
return t;
};
P = () => {
if (isUnary(next())) {
const op = unary(next());
consume();
const q = prec(op);
const t = Exp(q);
return { node: op, child: [t] };
} else if (next() === "LeftParen") {
consume();
const t = Exp(0);
expect("RightParen");
return t;
} else if (isTerminal(next())) {
const t = { node: next(), value: value() };
consume();
return t;
} else throw "SyntaxError: ";
};
return Eparser();
};
}
function explicit_formula(z1, z2, z3, z4) {
let f = (p, q) => {
if (p && q) {
return Boolean(z1);
} else if (p & !q) {
return Boolean(z2);
} else if (!p & q) {
return Boolean(z3);
} else if (!p & !q) {
return Boolean(z4);
}
};
return f;
}
return truthTable;
}
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