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<h1 class="title">Lecture – Rigidity Theory for Frameworks and Polytopes</h1>
</div>
<div class="quarto-title-meta">
</div>
</header>
<!--I will teach a 16 week course on *"Rigidity Theory for Frameworks and Polytopes"* at TU Berlin.
The course will run from October 15th until February 11th. -->
<p><img src="img/Teaching/flex.svg" width="270px" style="float:right; margin-top:-1em; margin-bottom:1em"></p>
<table>
<tbody><tr valign="top">
<td>
<strong>When</strong>
</td>
<td>
winter semester 2024/25 <br> every Tuesday, slot 10am – 12am (we meet 10:15) <br> October 15th, 2024 – February 11th, 2025
</td>
</tr>
<tr valign="top">
<td>
<strong>Where</strong><span class="math inline">\(\;\;\)</span>
</td>
<td>
TU Berlin <br> MA 751 (Charlottenburg)
</td>
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<!--<tr>
<td>**First lecture**$\quad$</td>
<td>October 15th, 2024</td>
</tr>
<tr>
<td>**Last lecture**</td>
<td>February 11th, 2025</td>
</tr>-->
</tbody></table>
<!-- https://befragung.tu-berlin.de/evasys/public/online/index/index?online_php=&pswd=WDNWD&ONLINEID=88788216653703123197813128729844126214016 -->
<p>You can find information about the course on <a href="https://isis.tu-berlin.de/course/view.php?id=41076">ISIS</a> and on <a href="https://moseskonto.tu-berlin.de/moses/verzeichnis/veranstaltungen/veranstaltung.html?veranstaltung=207713">Moses</a>.</p>
<section id="description" class="level3">
<h3 class="anchored" data-anchor-id="description">Description</h3>
<p><em>Rigidity</em> is a classical topic inspired from physics, engineering and architecture; but at the same time full of questions with intrinsically mathematical appeal and beauty.</p>
<p>Classical rigidity theory studies <em>frameworks</em> which are graphs that are embedded in Euclidean space with straight-line edges. One should think of them physically as built from rigid metal rods that are connected at universal joints. The central question of rigidity theory is whether a given framework is <em>flexible</em> (it can be deformed in such a way that all its edges stay of the same length) or <em>rigid</em> (it cannot be deformed in this way). Questions of this nature have a long history in structural engineering, but are also surprisingly ubiquitous in pure mathematics. The question for rigidity turns out intricat, and so many tools have been developed to deal with it, either approximately or in special case. Today mathematicians study many different forms of rigidity (infinitesimal, global, universal, generic, minimal, …) and many different settings for rigidity (bar-joint frameworks, point-hyperplane frameworks, volume rigidity, but also rigidity of manifolds and polytopes). The subject is intrinsically geometric, using tools from and inspiring results in algebraic geometry, projective geometry, hyperbolic geometry and convex geometry.</p>
<p>In this lecture we start with an introduction to the classical rigidity theory of frameworks which still underlies all modern developments. Beginning from the core definitions we explore the first- and second-order theory. We will take a look at different forms and setting for rigidity, such global and generic rigidity, and also point-hyperplane frameworks and some other forms of rigidity. In the second part of the lecture we focus specifically on rigidity questions that arise in the geometric study of <em>polytopes</em>. Here we take a look at classical rigidity results by Cauchy, Dehn, Gluck, Alexandrov and Minkowski, but also modern developments and their connection to <em>Wachspress Geometry</em>.</p>
</section>
<section id="scope" class="level3">
<h3 class="anchored" data-anchor-id="scope">Scope</h3>
<p>The first half of the lecture (roughly week 1 – 8) will focus on the classical rigidity theory of frameworks. The second half (roughly week 9 – 16) will explore the rigidity theory of polytopes. My goal is to talk about the following topics:</p>
<p><strong>Frameworks</strong></p>
<ul>
<li>fundamental concept: frameworks, tensegrities, flexes, rigidity (local, global, universal)</li>
<li>first-order theory: infinitesimal flexes, stresses, rigidity matrix, projective invariance</li>
<li>second-order theory: stress matrix, second-order rigidity, prestress stability, energy interpretation</li>
<li>generic rigidity, Maxwell counting condition, rigidity matroid , Laman graphs</li>
<li>global rigidity and generic global rigidity</li>
<li>frameworks on the sphere, cylinder etc.</li>
<li>cross-bracing theorems</li>
<li>point-hyperplane frameworks, volume rigidity</li>
<li>symmetry-forced rigidity, reduced counts, pure symmetry rigidity</li>
</ul>
<p><strong>Polytopes</strong></p>
<ul>
<li>rigidity theorems of Cauchy, Dehn and Gluck; Connelly’s flexible spheres, bellow theorem</li>
<li>polyhedral frameworks, Tutte embeddings, reciprocal frameworks, the Maxwell-Cremona correspondece</li>
<li>rigidity of triangulated surfaces</li>
<li>rigidity of non-triangulated surfaces with coplanarity constraints</li>
<li>coned polytope frameworks, the Wachspress-Izmestiev stress, stress-flex conjecture</li>
<li>uniqueness theorem’s of Alexandrov, Minkowski, etc.</li>
</ul>
</section>
<section id="prerequisites" class="level3">
<h3 class="anchored" data-anchor-id="prerequisites">Prerequisites</h3>
<p>The prerequisits are minimal. I will assume a good grasp of linear algebra and some fundamental graph theory (connectivitiy, planarity, handshaking lemma, etc). For the second part of the lecture some background knowledge on polytopes (definition, combinatorics) can be advantageous, but we will also recall the essentials.</p>
</section>
<section id="selected-literature" class="level3">
<h3 class="anchored" data-anchor-id="selected-literature">Selected Literature</h3>
<p>I list here some literature that was used in the preparation of the lecture and that might also be helpful to the interested student as an additional reading source.</p>
<p><strong>Infinitesimal rigidity and projective transformations</strong></p>
<ul>
<li>I. Izmestiev, <a href="https://link.springer.com/article/10.1007/s10711-008-9339-9"><em>“Projective Background of the Infinitesimal Rigidity of Frameworks”</em></a></li>
</ul>
<p><strong>Second-order rigidity</strong></p>
<ul>
<li>R. Connelly, W. Whiteley, <a href="https://pi.math.cornell.edu/~connelly/pdf/10.1137_S0895480192229236.pdf"><em>“Second-Order Rigidity and Prestress Stability for Tensegrity Frameworks”</em></a></li>
<li>R. Connelly, H. Servatius, <a href="https://link.springer.com/content/pdf/10.1007/BF02574003.pdf"><em>“Higher-Order Rigidity – What Is the Proper Definition?”</em></a></li>
</ul>
<p><strong>Rigidity of polytopes and polyhedra</strong></p>
<!--A large part of the material in the polytope section of the lecture is from Igor Pak's excellent book. A PDF version of the book is freely available on his website:-->
<ul>
<li>I. Pak, <a href="https://www.math.ucla.edu/~pak/book.htm"><em>“Lectures on Discrete and Polyhedral Geometry”</em></a> (in particular Chapter 26+)</li>
</ul>
<p><strong>Tutte embeddings and the Maxwell-Cremona correspondence</strong></p>
<p>There is a great course on Graph Theory by Don Sheehy available on YouTube that discusses these topics. Here are links to the relevant videos in the relevant order:</p>
<ul>
<li>Laplacians and Electricity [<a href="https://www.youtube.com/watch?v=M4f-fDOZByg">video</a>]</li>
<li>Solving Sub-Laplacian Systems [<a href="https://www.youtube.com/watch?v=AnnZypKlhjg">video</a>] <!--* Spectral Embedding [[video](https://www.youtube.com/watch?v=PcOI9hxYoj0)]--></li>
<li>Tutte’s Algorithm part 1 [<a href="https://www.youtube.com/watch?v=LzqiL3HbZPQ">video</a>]</li>
<li>Tutte’s Algorithm part 2 [<a href="https://www.youtube.com/watch?v=1J-_wFX49f0">video</a>]</li>
<li>Orienting the Boundary Matrix [<a href="https://www.youtube.com/watch?v=xbfoz1LKob0">video</a>]</li>
<li>The Maxwell-Cremona Correspondence part 1 [<a href="https://www.youtube.com/watch?v=kd1pS5bPdSI">video</a>]</li>
<li>The Maxwell-Cremona Correspondence part 2 [<a href="https://www.youtube.com/watch?v=WMPpOHRYUOw">video</a>]</li>
</ul>
</section>
<section id="lecture-material" class="level3">
<h3 class="anchored" data-anchor-id="lecture-material">Lecture material</h3>
<p>I do not intend to provide lecture notes consistently. The material that was made available online can be found here.</p>
<ul>
<li>0 - Introduction (<a href="teaching/rigidity-theory/0-Introduction.pdf">notes</a>)</li>
<li>1 - Frameworks & Rigidity (<a href="teaching/rigidity-theory/1-Frameworks & Rigidity.pdf">notes</a>)</li>
<li>I wrote a <a href="blog-posts/coning-frameworks/index.html">blog post</a> on radial moves in coned frameworks.</li>
</ul>
</section>
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arrow: false,
appendTo: function(el) {
return el.parentElement;
},
interactive: true,
interactiveBorder: 10,
theme: 'quarto',
placement: 'bottom-start',
};
if (contentFn) {
config.content = contentFn;
}
if (onTriggerFn) {
config.onTrigger = onTriggerFn;
}
if (onUntriggerFn) {
config.onUntrigger = onUntriggerFn;
}
window.tippy(el, config);
}
const noterefs = window.document.querySelectorAll('a[role="doc-noteref"]');
for (var i=0; i<noterefs.length; i++) {
const ref = noterefs[i];
tippyHover(ref, function() {
// use id or data attribute instead here
let href = ref.getAttribute('data-footnote-href') || ref.getAttribute('href');
try { href = new URL(href).hash; } catch {}
const id = href.replace(/^#\/?/, "");
const note = window.document.getElementById(id);
if (note) {
return note.innerHTML;
} else {
return "";
}
});
}
const xrefs = window.document.querySelectorAll('a.quarto-xref');
const processXRef = (id, note) => {
// Strip column container classes
const stripColumnClz = (el) => {
el.classList.remove("page-full", "page-columns");
if (el.children) {
for (const child of el.children) {
stripColumnClz(child);
}
}
}
stripColumnClz(note)
if (id === null || id.startsWith('sec-')) {
// Special case sections, only their first couple elements
const container = document.createElement("div");
if (note.children && note.children.length > 2) {
container.appendChild(note.children[0].cloneNode(true));
for (let i = 1; i < note.children.length; i++) {
const child = note.children[i];
if (child.tagName === "P" && child.innerText === "") {
continue;
} else {
container.appendChild(child.cloneNode(true));
break;
}
}
if (window.Quarto?.typesetMath) {
window.Quarto.typesetMath(container);
}
return container.innerHTML
} else {
if (window.Quarto?.typesetMath) {
window.Quarto.typesetMath(note);
}
return note.innerHTML;
}
} else {
// Remove any anchor links if they are present
const anchorLink = note.querySelector('a.anchorjs-link');
if (anchorLink) {
anchorLink.remove();
}
if (window.Quarto?.typesetMath) {
window.Quarto.typesetMath(note);
}
// TODO in 1.5, we should make sure this works without a callout special case
if (note.classList.contains("callout")) {
return note.outerHTML;
} else {
return note.innerHTML;
}
}
}
for (var i=0; i<xrefs.length; i++) {
const xref = xrefs[i];
tippyHover(xref, undefined, function(instance) {
instance.disable();
let url = xref.getAttribute('href');
let hash = undefined;
if (url.startsWith('#')) {
hash = url;
} else {
try { hash = new URL(url).hash; } catch {}
}
if (hash) {
const id = hash.replace(/^#\/?/, "");
const note = window.document.getElementById(id);
if (note !== null) {
try {
const html = processXRef(id, note.cloneNode(true));
instance.setContent(html);
} finally {
instance.enable();
instance.show();
}
} else {
// See if we can fetch this
fetch(url.split('#')[0])
.then(res => res.text())
.then(html => {
const parser = new DOMParser();
const htmlDoc = parser.parseFromString(html, "text/html");
const note = htmlDoc.getElementById(id);
if (note !== null) {
const html = processXRef(id, note);
instance.setContent(html);
}
}).finally(() => {
instance.enable();
instance.show();
});
}
} else {
// See if we can fetch a full url (with no hash to target)
// This is a special case and we should probably do some content thinning / targeting
fetch(url)
.then(res => res.text())
.then(html => {
const parser = new DOMParser();
const htmlDoc = parser.parseFromString(html, "text/html");
const note = htmlDoc.querySelector('main.content');
if (note !== null) {
// This should only happen for chapter cross references
// (since there is no id in the URL)
// remove the first header
if (note.children.length > 0 && note.children[0].tagName === "HEADER") {
note.children[0].remove();
}
const html = processXRef(null, note);
instance.setContent(html);
}
}).finally(() => {
instance.enable();
instance.show();
});
}
}, function(instance) {
});
}
let selectedAnnoteEl;
const selectorForAnnotation = ( cell, annotation) => {
let cellAttr = 'data-code-cell="' + cell + '"';
let lineAttr = 'data-code-annotation="' + annotation + '"';
const selector = 'span[' + cellAttr + '][' + lineAttr + ']';
return selector;
}
const selectCodeLines = (annoteEl) => {
const doc = window.document;
const targetCell = annoteEl.getAttribute("data-target-cell");
const targetAnnotation = annoteEl.getAttribute("data-target-annotation");
const annoteSpan = window.document.querySelector(selectorForAnnotation(targetCell, targetAnnotation));
const lines = annoteSpan.getAttribute("data-code-lines").split(",");
const lineIds = lines.map((line) => {
return targetCell + "-" + line;
})
let top = null;
let height = null;
let parent = null;
if (lineIds.length > 0) {
//compute the position of the single el (top and bottom and make a div)
const el = window.document.getElementById(lineIds[0]);
top = el.offsetTop;
height = el.offsetHeight;
parent = el.parentElement.parentElement;
if (lineIds.length > 1) {
const lastEl = window.document.getElementById(lineIds[lineIds.length - 1]);
const bottom = lastEl.offsetTop + lastEl.offsetHeight;
height = bottom - top;
}
if (top !== null && height !== null && parent !== null) {
// cook up a div (if necessary) and position it
let div = window.document.getElementById("code-annotation-line-highlight");
if (div === null) {
div = window.document.createElement("div");
div.setAttribute("id", "code-annotation-line-highlight");
div.style.position = 'absolute';
parent.appendChild(div);
}
div.style.top = top - 2 + "px";
div.style.height = height + 4 + "px";
div.style.left = 0;
let gutterDiv = window.document.getElementById("code-annotation-line-highlight-gutter");
if (gutterDiv === null) {
gutterDiv = window.document.createElement("div");
gutterDiv.setAttribute("id", "code-annotation-line-highlight-gutter");
gutterDiv.style.position = 'absolute';
const codeCell = window.document.getElementById(targetCell);
const gutter = codeCell.querySelector('.code-annotation-gutter');
gutter.appendChild(gutterDiv);
}
gutterDiv.style.top = top - 2 + "px";
gutterDiv.style.height = height + 4 + "px";
}
selectedAnnoteEl = annoteEl;
}
};
const unselectCodeLines = () => {
const elementsIds = ["code-annotation-line-highlight", "code-annotation-line-highlight-gutter"];
elementsIds.forEach((elId) => {
const div = window.document.getElementById(elId);
if (div) {
div.remove();
}
});
selectedAnnoteEl = undefined;
};
// Handle positioning of the toggle
window.addEventListener(
"resize",
throttle(() => {
elRect = undefined;
if (selectedAnnoteEl) {
selectCodeLines(selectedAnnoteEl);
}
}, 10)
);
function throttle(fn, ms) {
let throttle = false;
let timer;
return (...args) => {
if(!throttle) { // first call gets through
fn.apply(this, args);
throttle = true;
} else { // all the others get throttled
if(timer) clearTimeout(timer); // cancel #2
timer = setTimeout(() => {
fn.apply(this, args);
timer = throttle = false;
}, ms);
}
};
}
// Attach click handler to the DT
const annoteDls = window.document.querySelectorAll('dt[data-target-cell]');
for (const annoteDlNode of annoteDls) {
annoteDlNode.addEventListener('click', (event) => {
const clickedEl = event.target;
if (clickedEl !== selectedAnnoteEl) {
unselectCodeLines();
const activeEl = window.document.querySelector('dt[data-target-cell].code-annotation-active');
if (activeEl) {
activeEl.classList.remove('code-annotation-active');
}
selectCodeLines(clickedEl);
clickedEl.classList.add('code-annotation-active');
} else {
// Unselect the line
unselectCodeLines();
clickedEl.classList.remove('code-annotation-active');
}
});
}
const findCites = (el) => {
const parentEl = el.parentElement;
if (parentEl) {
const cites = parentEl.dataset.cites;
if (cites) {
return {
el,
cites: cites.split(' ')
};
} else {
return findCites(el.parentElement)
}
} else {
return undefined;
}
};
var bibliorefs = window.document.querySelectorAll('a[role="doc-biblioref"]');
for (var i=0; i<bibliorefs.length; i++) {
const ref = bibliorefs[i];
const citeInfo = findCites(ref);
if (citeInfo) {
tippyHover(citeInfo.el, function() {
var popup = window.document.createElement('div');
citeInfo.cites.forEach(function(cite) {
var citeDiv = window.document.createElement('div');
citeDiv.classList.add('hanging-indent');
citeDiv.classList.add('csl-entry');
var biblioDiv = window.document.getElementById('ref-' + cite);
if (biblioDiv) {
citeDiv.innerHTML = biblioDiv.innerHTML;
}
popup.appendChild(citeDiv);
});
return popup.innerHTML;
});
}
}
});
</script>
</div> <!-- /content -->
</body></html>