Section 4: Patterns, Empirical Generalizations¶

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Six sessions of counting, pooling data, and asking whether a regularity is a law

Overview¶

The syllabus 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 the obvious rule holds case after case and then fails. In Neutrinos 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. 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, "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 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; 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.

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 in class; only its name survives, and the page reconstructs it as a lab in pooling counts. Deal with Platonic Solids and Applications, 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 the dealer invents a secret rule and the players discover it by experiment. Deal with The Mirror Mystery.

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 begins in this last session of Section 4 and continues into Section 5, 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 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).
• Stopping at confirmations. Five agreeing cases do not guarantee the sixth. Choose the test most likely to break the rule (Eleusis 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 asks you to build the solid where the rule fails).
• Believing what everyone believes. Vary the observation before explaining it (The Mirror Mystery).
• Biased samples. One specimen, or one group's counts, is an anecdote. Pool the data (Cell Shapes Lab, Egg Pouches Lab).

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). An empirical law can be right long before anyone can say why, as Kepler's rule was (Paired Observations).

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; Internet Archive 🔓 (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; publisher 🔒
• Robert Ehrlich, Nine Crazy Ideas in Science, Chapter 6: "The Solar System Has Two Suns" (session 21). Reading list entry; publisher 🔒
• James L. Adams, Conceptual Blockbusting, Chapter 6: Alternative thinking languages (session 22). Reading list entry; publisher 🔒
• Robert Ehrlich, Nine Crazy Ideas in Science, Chapter 7: "Oil, Coal, and Gas Have Abiogenic Origins" (session 23). Reading list entry; publisher 🔒
• Robert Ehrlich, Nine Crazy Ideas in Science, Chapter 10: "There Was No Big Bang" (session 24). Reading list entry; publisher 🔒

Problems in this section¶

See the problem index 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.

Machine-readable: this page as Markdown · raw source on GitHub · llms.txt · llms-full.txt (whole site). See For AI agents.