--- title: "Section 4: Patterns, Empirical Generalizations" description: "Sessions 19 to 24 of Winfree's course: finding regularities in counts, tables and games, pooling class data, and telling a pattern observed from a pattern explained, with the Cell Shapes lab, Eleusis, and the start of the Stacked Cantilevers lab." type: Lesson tags: [course, student-facing, section-4, patterns, generalization, data] status: stable generated: by: "claude/opus-5" at: "2026-09-16T00:00:00Z" sources: - id: winfree-handout resource: "https://web.archive.org/web/20070214070741/http://eebweb.arizona.edu/faculty/winfree/Handout_479.htm" title: "The Art of Scientific Discovery (EEB 479): course handout, Web Archive capture of 2007-02-14" author: "Arthur T. Winfree" - id: aosd-syllabus resource: "https://github.com/tyson-swetnam/aosd/blob/main/docs/assets/aosd_syllabus.pdf" title: "The Art of Scientific Discovery: original course syllabus (PDF)" author: "Arthur T. Winfree" --- # Section 4: Patterns, Empirical Generalizations

This work is licensed under a Creative Commons Attribution 4.0 International License. *Six sessions of counting, pooling data, and asking whether a regularity is a law* ## Overview The [syllabus](syllabus.md) calls this section "Patterns, Empirical Generalizations". Sections 1 to 3 taught you to check facts, notice your blocks and ask good questions. Now the facts pile up, and the job is to find the rule inside them: in chords drawn across a circle, a table of paired numbers, a patch of cells, a row of accepted playing cards. A rule that fits the data is not yet a rule you understand. In [N Dots on the Rim of a Circle](problems/n-dots-on-circle.md) the obvious rule holds case after case and then fails. In [Neutrinos](problems/neutrinos.md) a pattern that most people expected to go away survived for thirty years. The syllabus calls its exercises practice in "cultivating multiple alternative solutions" and "eliminating rejectable candidate solutions". A pattern is a candidate like any other. Four activities happen in class: the Cell Shapes and Egg Pouches labs, the start of the Stacked Cantilevers lab, and a game of Eleusis. Some patterns only appear in more data than one person can gather. As the syllabus puts it: "Class meetings will also prove essential for some problems in which no one individual can collect enough data, but if we pool data, reality will come into focus." ## The sessions For most of these problems the syllabus gives only a name, and Winfree's problem sheets are lost. The problem pages are the editors' reconstructions, and each says how sure the identification is. ### Session 19: Pattern **Readings due:** Judson, Chapter 2: *Pattern*. Discuss [Presidents and States](problems/presidents-and-states.md). What Winfree meant by it is unknown; the page offers two regularities from American history that look like laws. Discuss [N Dots on the Rim of a Circle](problems/n-dots-on-circle.md), "connected to slice the disk": count the pieces, predict the next case, then check. ### Session 20: The Cell Shapes lab begins **Readings due:** Ehrlich, Chapter 5: "Low Doses of Nuclear Radiation Are Beneficial". "Start Cell Shapes lab in class." The [Cell Shapes Lab](problems/cell-shapes-lab.md) runs for three sessions. No lab sheet survives; the page suggests counting the sides of cells in a flat froth or a leaf peel first, and theorizing later. ### Session 21: Pooled experiments, paired numbers **Readings due:** Ehrlich, Chapter 6: "The Solar System Has Two Suns". "Collaborative experiments on Cell Shapes", the second lab session. Deal with [Paired Observations](problems/paired-observations.md); in a 2002 web copy of Winfree's handout this link points to a bookmark named `Keplers_Laws`, the clue the reconstruction follows. Deal with [Neutrinos](problems/neutrinos.md). ### Session 22: Alternative thinking languages **Readings due:** Adams, Chapter 6: Alternative thinking languages. "Further experiments on cell shapes", the lab's last scheduled session. Do the [Egg Pouches Lab](problems/egg-pouches-lab.md) in class; only its name survives, and the page reconstructs it as a lab in pooling counts. Deal with [Platonic Solids and Applications](problems/platonic-solids.md), where a sketch or a straw model does work that words cannot. ### Session 23: Eleusis **Readings due:** Ehrlich, Chapter 7: "Oil, Coal, and Gas Have Abiogenic Origins". "Play Eleusis in class": in [Eleusis](problems/eleusis.md) the dealer invents a secret rule and the players discover it by experiment. Deal with [The Mirror Mystery](problems/mirror-mystery.md). ### Session 24: The Stacked Cantilevers lab begins **Readings due:** Ehrlich, Chapter 10: "There Was No Big Bang". "Start Stacked Cantilevers lab." The [Stacked Cantilevers Lab](problems/stacked-cantilevers-lab.md) begins in this last session of Section 4 and continues into [Section 5](section5.md), where it is listed with that section's problems: "Further collaborations" in session 25 and "Theory of stacking cantilevers, resolution of wagers" in session 26. ## Key ideas **Observed is not explained.** A rule that fits every case you have drawn is still a guess until you can say why it must hold. [N Dots on the Rim of a Circle](problems/n-dots-on-circle.md) is the classic warning; session 4's phrase "distinguishing things we know vs only imagine" applies to patterns too. **How pattern-finding goes wrong.** - *Seeing patterns in noise.* Search enough rules and some will fit by chance. Ask how many others you could have tried ([Presidents and States](problems/presidents-and-states.md)). - *Stopping at confirmations.* Five agreeing cases do not guarantee the sixth. Choose the test most likely to break the rule ([Eleusis](problems/eleusis.md) answers only right or wrong, so the card you choose is the experiment). - *Overgeneralizing.* A rule true in its home range can fail outside it. Look for the boundary ([Platonic Solids and Applications](problems/platonic-solids.md) asks you to build the solid where the rule fails). - *Believing what everyone believes.* Vary the observation before explaining it ([The Mirror Mystery](problems/mirror-mystery.md)). - *Biased samples.* One specimen, or one group's counts, is an anecdote. Pool the data ([Cell Shapes Lab](problems/cell-shapes-lab.md), [Egg Pouches Lab](problems/egg-pouches-lab.md)). **Some patterns survive every attack.** The solar neutrino shortfall held for thirty years while most people assumed the calculation or the detectors were wrong ([Neutrinos](problems/neutrinos.md)). An empirical law can be right long before anyone can say why, as Kepler's rule was ([Paired Observations](problems/paired-observations.md)). **Separate what is forced from what is only usual.** In a froth some regularities follow from topology and some are merely typical. Counting first, then proving what you can, tells them apart. ## GamesWorth focus for this section - Before drawing or counting the next case, write your prediction down with the date and time, then record whether it held. - Keep a list of every rule you tried and discarded, not just the one that survived. The syllabus asks you to record "how you got into and out of blind alleys." - After each Eleusis round, log each hypothesis and the card you played to test it. - For every generalization, add one line: *what observation would break this?* - On a left-hand page, write a morning-after on N Dots on the Rim of a Circle or Paired Observations: when did you first believe the rule, and on what evidence? ## Readings for this section - **Horace Freeland Judson**, *The Search for Solutions*, Chapter 2: *Pattern* (session 19). [Reading list entry](readings.md#judson); [Internet Archive](https://archive.org/details/searchforsolutio00juds){target=_blank} 🔓 *(borrow)* - **Robert Ehrlich**, *Nine Crazy Ideas in Science: A Few Might Even Be True*, Chapter 5: "Low Doses of Nuclear Radiation Are Beneficial" (session 20). [Reading list entry](readings.md#ehrlich); [publisher](https://press.princeton.edu/books/paperback/9780691094953/nine-crazy-ideas-in-science){target=_blank} 🔒 - **Robert Ehrlich**, *Nine Crazy Ideas in Science*, Chapter 6: "The Solar System Has Two Suns" (session 21). [Reading list entry](readings.md#ehrlich); [publisher](https://press.princeton.edu/books/paperback/9780691094953/nine-crazy-ideas-in-science){target=_blank} 🔒 - **James L. Adams**, *Conceptual Blockbusting*, Chapter 6: Alternative thinking languages (session 22). [Reading list entry](readings.md#adams); [publisher](https://www.hachettebookgroup.com/titles/james-l-adams/conceptual-blockbusting/9781541674059/?lens=basic-books){target=_blank} 🔒 - **Robert Ehrlich**, *Nine Crazy Ideas in Science*, Chapter 7: "Oil, Coal, and Gas Have Abiogenic Origins" (session 23). [Reading list entry](readings.md#ehrlich); [publisher](https://press.princeton.edu/books/paperback/9780691094953/nine-crazy-ideas-in-science){target=_blank} 🔒 - **Robert Ehrlich**, *Nine Crazy Ideas in Science*, Chapter 10: "There Was No Big Bang" (session 24). [Reading list entry](readings.md#ehrlich); [publisher](https://press.princeton.edu/books/paperback/9780691094953/nine-crazy-ideas-in-science){target=_blank} 🔒 ## Problems in this section | Session | Problem | Kind | What it trains | | :-- | :-- | :-- | :-- | | 19 | [Presidents and States](problems/presidents-and-states.md) | discussion | Pattern, accident or law? (editors' reconstruction) | | 19 | [N Dots on the Rim of a Circle](problems/n-dots-on-circle.md) | puzzle | Pattern observed vs pattern explained | | 20-22 | [Cell Shapes Lab](problems/cell-shapes-lab.md) | lab | Counting and pooling before theorizing | | 21 | [Paired Observations](problems/paired-observations.md) | puzzle | Finding a law in a table, then testing it | | 21 | [Neutrinos](problems/neutrinos.md) | case study | A pattern that would not go away | | 22 | [Egg Pouches Lab](problems/egg-pouches-lab.md) | lab | Generalizing from pooled specimens (editors' reconstruction) | | 22 | [Platonic Solids and Applications](problems/platonic-solids.md) | puzzle | Finding a rule's limits, then proving it | | 23 | [Eleusis](problems/eleusis.md) | puzzle | Induction by experiment | | 23 | [The Mirror Mystery](problems/mirror-mystery.md) | puzzle | Checking a belief everyone shares | See the [problem index](problems/index.md) for every problem in the course. --- *A pattern is where an explanation starts, not where it ends: count, predict, try to break it, and only then ask why it holds.*