# Administrative Information - **Class Meeting**: Friday, September 18, 2026, 1:00–1:50pm — **Coolbaugh 212** - Previous meeting: [[MATH310F26(Day 9) - The rectangular pulse]] - 📄 [[Teaching/MATH310/MATH310F26/Administration/MATH310_F26_Syllabus_1.0.pdf|MATH 310 F26 Syllabus, v1.0]] > [!info]- Admin. Notes > Expect an LR and WB for each lecture. WBs target 15-minute engagements (longer early on); LRs are short, ~5 minutes. > > - 🎯 **[LR] [Day 10 Reflection Questions](https://elearning.mines.edu/courses/81632/quizzes/121340)** — 4 questions drawn from today's material, unlimited attempts, highest kept — due **Monday 9/21, 11:59pm** > - 🎯 **[LR] [Day 10 Reflection Statement](https://elearning.mines.edu/courses/81632/quizzes/121341)** — one free response, no wrong answers — due **Monday 9/21, 11:59pm** > - 🎯 **[LR] [Day 9 Reflection Questions](https://elearning.mines.edu/courses/81632/quizzes/121253)** — 4 questions drawn from Monday's material, unlimited attempts, highest kept — due **Friday 9/18, 11:59pm** > - 🎯 **[LR] [Day 9 Reflection Statement](https://elearning.mines.edu/courses/81632/quizzes/121254)** — one free response, no wrong answers — due **Friday 9/18, 11:59pm** > - 🎯 **[WB] [Workbook 6](https://elearning.mines.edu/courses/81632/assignments/586618)** · [[MATH310F26_WB6.pdf|blank]] · [[MATH310F26(Workbook 6 Companion) - What a conjugate pair extracts|companion]] > - 🎯 **[WB] [Workbook 7](https://elearning.mines.edu/courses/81632/assignments/586619)** · [[MATH310F26_WB7.pdf|blank]] · [[MATH310F26(Workbook 7 Companion) - Two tones and a constant, transformed|companion]] > - 🎯 **[WB] [Workbook 8](https://elearning.mines.edu/courses/81632/assignments/587522)** — a constant meets its Fourier series (today's Example 1: graph, the orthogonality integral, $c_0=2\pi$); upload one PDF > - 📄 [[MATH310F26_WB8.pdf|Workbook 8 (blank, 1 page)]] > - Start with the companion: [[MATH310F26(Workbook 8 Companion) - One tone, and the integral that finds it]] > - 🎯 **[WB] [Workbook 9](https://elearning.mines.edu/courses/81632/assignments/587523)** — from complex tones to sines (today's Example 2: the algebraic reduction to $\sum 2(-1)^{n+1}\sin(nx)/n$); upload one PDF > - 📄 [[MATH310F26_WB9.pdf|Workbook 9 (blank, 1 page)]] > - Start with the companion: [[MATH310F26(Workbook 9 Companion) - The even reduction]] > - 🎯 **[MP] One Question, and the First Arrow** — one question, your review-or-project choice, and the first step of the modeling cycle; type your responses in the [Canvas quiz](https://elearning.mines.edu/courses/81632/quizzes/121327), **or** print the two-sided form ([[MATH310F26-OneQuestionFirstArrow.pdf|blank PDF]]) and attach a scan — read [[MATH310F26 - The Modeling Cycle (Part 1)]] first — due **Friday 9/25, 11:59pm** > - Meanwhile: the [[Fourier Series]] and [[Fourier Transforms]] applet pages are live — an assignment built on them is coming > [!example]- 🗃️ COMAP Corner — three from the archive > Three from the archive — one data, one computational, one weird — for a day of listening: > > - **Data/stats — Searching for a Lost Plane (2015 MCM B).** Inspired by MH370: a plane goes down in open ocean with no emergency signal. Build a search strategy that weighs sensor capabilities and generalizes across oceans and routes — probability doing rescue work. 📄 [[2015_MCM_Problem_A_Eradicating_Ebola_Problem_B_Searching_for_a_lost_plane.html|problem page (archive)]] · [comap.com](https://www.contest.comap.com/undergraduate/contests/mcm/contests/2015/problems/) > - **Computational — Repeater Coordination (2011 MCM B).** VHF repeaters extend low-power radios but interfere unless separated in space or frequency. How few repeaters serve 1,000 simultaneous users in a 40-mile radius — and what breaks at 10,000? Spectrum as a scarce resource. 📄 [[2011_MCM_Problem_A_Snowboard_Course_Problem_B_Repeater_Coordination.html|problem page (archive)]] · [comap.com](https://www.contest.comap.com/undergraduate/contests/mcm/contests/2011/problems/) > - **Just weird — The Drug Runner Problem (1988 MCM A).** Two shore listening posts catch one brief radio signal from a boat waiting offshore, each bearing good to only 2°. Bound where the boat can be, then design the helicopter's search pattern — 1988's idea of signals intelligence. 📄 [[1988_MCM_Problem_A_The_Drug_Runner_Problem.pdf|problem PDF]] · [comap.com](https://www.contest.comap.com/undergraduate/contests/matrix/PDF/1988/1988A.pdf) --- # Deliverables - 🎯 **[LR] [Day 10 Reflection Questions](https://elearning.mines.edu/courses/81632/quizzes/121340)** — 4 questions drawn from today's material, unlimited attempts, highest kept - 🎯 **[LR] [Day 10 Reflection Statement](https://elearning.mines.edu/courses/81632/quizzes/121341)** — one free response, no wrong answers - 🎯 **[WB] [Workbook 8](https://elearning.mines.edu/courses/81632/assignments/587522)** — a constant and its Fourier series; upload one PDF - 🎯 **[WB] [Workbook 9](https://elearning.mines.edu/courses/81632/assignments/587523)** — from complex modes to sinusoids; upload one PDF - 🎯 **[MP] [One Question, and the First Arrow](https://elearning.mines.edu/courses/81632/quizzes/121327)** — due Friday 9/25 at 11:59 pm Questions and statement are due Monday 9/21 at 11:59 pm. The Day 9 pair closes tonight, Friday 9/18 at 11:59 pm. --- # Lecture Boards + Transcript + GenAI **Tenth meeting, September 18 — hearing the Fourier transform.** Hope career fair was good, or at least the swag was useful. The day's mission statement: start incorporating the mathematics into *sense organs* — Monday we saw Fourier, today we hear it. And the arc ahead: Fourier transform transitioning into **Fourier series** today, the suppressed derivation on Monday, modeling questions through the week, and **stochastic processes** waiting after Fourier. ## Housekeeping **Modeling Project Activity #1 is live** — a page on the site and a Canvas questionnaire (or the one-page front-and-back form, your choice), taking the questions you've been asking and pushing them toward *let's get started*: due **next Friday 9/25** (links in the Admin callout above). Nothing is set in stone — pivoting stays legal — but scaffolding is going up, with a concrete target: in about **four weeks**, a *crude* mathematical pipeline connecting what you're thinking contextually to what the math is saying. Crude is a feature: brainstorming runs broad and complicated, but the first mathematical pass should be as simple as possible, so the math can start talking back and iteration can begin off a functional base. (The page exists because SIAM published a book on modeling practice whose famous diagram — two rows of three circles, arrows interconnecting *literally all of them* — is not wrong, but reads as "what am I supposed to do?"; the page offers one instructor's actual path through it.) Indecision or brainstorming: office hours, or one-on-one appointments, which are usually the efficient version. **Next Friday is an activity day.** The course TA — a former student of this course — has agreed to give a mock 12-minute presentation on his project from a year and a half ago, pitched *metacognitively*: the steps, the hurdles, the thought process, not the technical detail. Asked whether the instructor should give one too, the room voted for both. "Diversity? OK, we'll go diversity. I don't know what I'm going to talk about. But I've got a week." **Exhaustion poll:** 1 to 10, tens surfaced instantly. Accordingly, today has board math, a demonstration device, and things to listen to. > [!tip]- The bots aligned themselves with Squiddy (and answered Lily's question) > A reveal from the site: after Monday's class, the transcript and boards went in for the usual rebuild, and the bots — unprompted — **put the eyes on Squiddy**, in the note's figure *and* inside the applet, where the eyes move. "Big ups to the bots for being aligned with content there." (A sharp-eyed student also found the applet's get-serious button; noted, and now you all know.) More substantively, the follow-up computation from Monday's \$10 question came up: wedges of Fourier-space energy kept, everything else deleted, and the inverse transform run to paint *which pixels* each wedge came from — remember that $\hat f(\omega)$ consolidates **all** of the original data, so a region of frequency space can light up regions scattered anywhere in the image. The wedge Lily asked about traces to the tower-like structures; the fine outer wedges to the little line details. And a spooky consequence of that all-at-once character: **the Fourier transform owes causality nothing.** Compress applause aggressively as MP3 — a pure frequency-space game of chuck-data-to-compress — and listen back: there is a *pre-echo* before the first clap. Something sounds before anything officially happened, because frequency space gathered up all of time before answering. ## From the transform to a series: ideal localization, en masse Both transforms on the board, as always — data to amplitude-per-frequency, and amplitude-per-frequency interfered back into data. Now the move of the day. **Suppose we decide to ideally localize amplitude in frequency space** — not at one frequency, but at a whole *countable* family $\omega_{n}\in\mathbb{R}$, $n\in\mathbb{Z}$ (the funky $\mathbb{Z}$: integers, the fingers-and-toes numbers with nothingness and the negatives): $\hat f(\omega)=\sum_{n=-\infty}^{\infty}c_{n}\,\delta(\omega-\omega_{n}),\qquad c_{n}\in\mathbb{C}.$ Feed this comb to the inverse transform (spatial variable $x$ today): $f(x)=\frac{1}{2\pi}\int_{-\infty}^{\infty}\hat f(\omega)\,e^{i\omega x}\,d\omega=\frac{1}{2\pi}\int_{-\infty}^{\infty}\sum_{n=-\infty}^{\infty}c_{n}\,\delta(\omega-\omega_{n})\,e^{i\omega x}\,d\omega.$ Two hats to wear at the sum-integral swap. The **theoretical hat** notes there are two limits here — the countable sum and the continuous one — and swapping them is exactly the kind of move that wants uniform convergence before it's legal. The **applied hat** says: fiddle with the things and see what happens; if they tell you good things, backtrack and rigorously justify *if that's your game* — and if they were never going to tell you anything useful, no reason to ask theoretical questions about them. Applied hat on. Swap, pull out the constants, and let each delta do the one thing deltas do inside integrals (sift; total area one): $f(x)=\frac{1}{2\pi}\sum_{n=-\infty}^{\infty}c_{n}\int_{-\infty}^{\infty}\delta(\omega-\omega_{n})\,e^{i\omega x}\,d\omega=\frac{1}{2\pi}\sum_{n=-\infty}^{\infty}c_{n}\,e^{i\omega_{n}x}.\qquad(1)$ The continuous superposition has collapsed into a $\Sigma$-style sum: a function *sourced by* ideally localized frequencies. ## One base frequency, and periodicity falls out So far the $\omega_{n}$ are unrelated. Now give them structure: $\omega_{n}=n\pi/L$ for some $L>0$ — one important base frequency $\pi/L$, everything else an integer multiple of it. (Unrelated frequencies would crank up the propensity for noisy things; related ones might make something interesting happen.) Test Eq. (1) for a *horizontal shift* by $2L$: everywhere an $x$, put $x+2L$, and the knee-jerk for addition in an exponent splits the exponential: $e^{i\omega_{n}(x+2L)}=e^{i\omega_{n}x}\,e^{i\frac{n\pi}{L}\cdot 2L}=e^{i\omega_{n}x}\,e^{2\pi n i}.$ And what is $e^{2\pi ni}$? Cosine and sine of full laps around the unit circle: **one.** (The room split briefly on $\sin(4\pi)$; the unit circle settled it.) So every trace of the shift evaporates: $f(x+2L)=f(x)$ — **Eq. (1) is a $2L$-periodic function, and is called a Fourier series representation of $f$.** The distributional machinery bought us the classical object. And a classical theorem catches us: if $\int_{-L}^{L}|f(x)|^{2}\,dx<\infty$, the series converges to $f$ **in the least-squares sense** — from another vantage point this whole construction is one more least-squares process, which is not today's point but is a very course-shaped fact to file. Why care about periodic functions? Because phenomena do. A ukulele appeared ("seems out of tune, right?"): pluck the string, the string vibrates the air, the air vibrates the eardrum, the eardrum shakes cilia tuned to resonant frequencies, kinetic energy trades to electrical, and that is hearing. A plucked shape on $[0,L]$ — a teepee, a bow — is not itself periodic; the mathematics says *tile it*: copy the shape to $[L,2L]$, $[2L,3L]$, forever both ways, and the tiled function **is** periodic, hence owns a well-defined series representation, hence everything on the string has understood structure underneath. ## The sawtooth, with the derivation suppressed Example: let $f(x)=x$ for $x\in(-\pi,\pi)$, extended by $f(x+2\pi)=f(x)$ — the line $y=x$, restricted, then repeated, with jump discontinuities at every seam: the **sawtooth wave**. (One board copy briefly cohabitated orange-and-blue, Broncos country; redrawn.) At 35 minutes past, with the room voting to *hear things* before the weekend, the derivation got the infomercial treatment — from a derivation (**see Monday**), fully cooked and pulled from under the table: $f(x)=\sum_{n=1}^{\infty}\frac{2(-1)^{n+1}}{n}\,\sin(nx).$ Come for the hearing, stay for Monday's derivation. Read it as a recipe: $\sin(x)$ repeats every $2\pi$; $\sin(2x)$ twice as often; the amplitudes $2(-1)^{n+1}/n$ are rigged *just right* so that when the waves lay on top of each other, the parts that look like $y=x$ amplify and the parts that don't deconstruct: ![[day10_sawtooth_sums.png]] ## Hearing it — the applets, in class and here The segue began with a Taylor-series applet for contrast: power functions splinting onto $\sin(x)$ from a point of tangency, McLaurin-style, convergence wrapping on term by term. Same game today, new base elements: **sinusoids instead of power functions**, correction terms that are *more frequent waves sharing the base frequency* rather than higher powers. Then the class watched the sawtooth build — mode one, plus two, plus three, four, five, each strip recorded on the left, the sum smooshing down onto $y=x$ on the right — and *heard* it: the **fundamental** alone (a pure boop), then all 16 terms at once. The room's verdict on the sawtooth: *icky*. The triangle wave, same experiment: much closer to its fundamental, much easier on the ear. The square: harsh again. The diagnosis is the jumps — **discontinuities create harshness**, because they force strong high harmonics; the harmonics are the *texture* on the fundamental. (Those side ripples you were told to ignore for now: noted, deliberately, for later — they have a name.) The ukulele confirmed the physics: touch — don't press — the middle of the string and pluck: the fundamental's shape is forbidden, the second mode sounds alone. Strings and flute-tubes alike allow only certain shapes, a bass tone plus higher harmonics adding character. Both instruments from class live on this site — play the presets, then draw your own wave and listen to it: <div class="applet" data-applet="fourier-waves"></div> <div class="applet" data-applet="fourier-draw"></div> The full guided page, with the coefficient formulas, Parseval-as-$R^2$, and things worth trying: [[Fourier Series]] (and the transform-side page: [[Fourier Transforms]]). ## 8-bit music, and the Chipophone Now step into electronic land, where the harshness was once a *feature*. Picture storing one period of a smooth sinusoid on 1980s hardware: tracing that curve takes a lot of data points. But a **triangle wave**? Save three corners and connect the dots. A **square wave**? Save *low* and *high*. If you are an 8-bit Nintendo Entertainment System, that is the whole game: the NES didn't store recordings of its music at all — its sound chip *generated* waves cheap enough to describe in a few numbers, which is why its channels are exactly the shapes from today's applet: two pulse/square channels, a triangle channel, and noise. Every jump discontinuity in those waves pours energy into high harmonics — the sawtooth's $1/n$-slow amplitude decay versus the smoother triangle's faster fade, audible ten minutes ago — and that bright, buzzy harmonic ladder **is** the chiptune sound. "So that's why it sounded like that back in the day. Now we can just give high-fidelity music on the game right away." And then the demonstration that has to be seen: a musician gutted an old electric organ and rebuilt it as a live-playable 8-bit synthesizer — square waves, the four-bit triangle ("triangles, a little better — easy on storage"), the flat flute, all under two hands:  > [!quote] Attribution > **"The Chipophone"** — Linus Åkesson (lftkryo): [youtube.com/watch?v=m1pchpDD5EU](https://www.youtube.com/watch?v=m1pchpDD5EU). A homemade 8-bit synthesizer in a repurposed electric organ; more at [linusakesson.net/chipophone](https://www.linusakesson.net/chipophone/index.php). One last tease before the weekend, delivered but not explored: the second applet above lets you **draw a shape and play that shape**. Go make an icky sound on purpose. Have a good weekend. > [!success]- Numbers checked in one sandbox run > The infomercial punchline was verified as a working recipe: partial sums of $\sum 2(-1)^{n+1}\sin(nx)/n$ at $x=1,\,2,\,-0.5$ ran $1.6829\to1.2461\to1.0499\to1.0030$ (and analogously at the other points) for $N=1,4,16,200$ — converging on $x$ itself, as the least-squares theorem promises for this square-integrable $f$. The periodicity engine returned $e^{i\omega_{n}\cdot 2L}=1$ to machine precision at $L=0.7$, $n=5$. And the ignored-for-now ripples were measured: $\max S_{200}\approx 3.688$ against the jump-edge value $\pi$, an overshoot of $\approx 8.7\%$ of the jump and climbing toward the applet readout's $9.0\%$ — a named phenomenon with its reveal reserved. > [!example]- Board scans — September 18, 2026 > ![[MATH310F26-Day10-Board-1.jpg]] > ![[MATH310F26-Day10-Board-2.jpg]] > ![[MATH310F26-Day10-Board-3.jpg]] > ![[MATH310F26-Day10-Board-4.jpg]] > > 📄 [[MATH310F26-Day10-Boards-2026-09-18.pdf|Full board capture (4 pages, PDF)]]