One read of Claude Documentationclaude-docs-20261016T160705Z

claude-science/example-gallery New page · 27 lines, new page

# Example gallery

A whole new page. There's nothing to diff it against, so here is what it says.

# Example gallery

> Browse nine example prompts from nine fields of science, each the start of a complete analysis in Claude Science.

Copy an example's prompt into a new session in [Claude Science](/docs/claude-science/overview) and compare your result with its reference analysis.

<div className="example-gallery-cards">
  <CardGroup cols={2}>
    <Card title="Genetic drift in a bottle: Buri's 107 fruit-fly populations" img="https://mintcdn.com/claude-ai/F2QP9ubtYgT0edFI/images/claude-science/example-gallery/bio-drift-card-light.webp?fit=max&auto=format&n=F2QP9ubtYgT0edFI&q=85&s=a5a85c0e38dda18a14c16f7d778a592f" href="/docs/claude-science/example-gallery/bio-drift" width="1200" height="800" data-path="images/claude-science/example-gallery/bio-drift-card-light.webp" />

    <Card title="A titration curve from the full equilibrium: which indicator finds the equivalence point?" img="https://mintcdn.com/claude-ai/F2QP9ubtYgT0edFI/images/claude-science/example-gallery/chem-titration-card-light.webp?fit=max&auto=format&n=F2QP9ubtYgT0edFI&q=85&s=170d2e89ce460c2563c0a3212645712b" href="/docs/claude-science/example-gallery/chem-titration" width="1200" height="800" data-path="images/claude-science/example-gallery/chem-titration-card-light.webp" />

    <Card title="A first Kaplan–Meier curve: the 1963 leukemia trial of 6-MP" img="https://mintcdn.com/claude-ai/F2QP9ubtYgT0edFI/images/claude-science/example-gallery/clin-km-card-light.webp?fit=max&auto=format&n=F2QP9ubtYgT0edFI&q=85&s=a2eda48c7c7f1ae4582384c50e901ef1" href="/docs/claude-science/example-gallery/clin-km" width="1200" height="800" data-path="images/claude-science/example-gallery/clin-km-card-light.webp" />

    <Card title="PageRank on a web of 11 pages: why 1 good link beats 6 weak ones" img="https://mintcdn.com/claude-ai/F2QP9ubtYgT0edFI/images/claude-science/example-gallery/cs-pagerank-tiny-card-light.webp?fit=max&auto=format&n=F2QP9ubtYgT0edFI&q=85&s=bd9dcdc46370ffab0c1db6d4efe14317" href="/docs/claude-science/example-gallery/cs-pagerank-tiny" width="1200" height="800" data-path="images/claude-science/example-gallery/cs-pagerank-tiny-card-light.webp" />

    <Card title="A first finite-element model: a clamped steel ruler bends and vibrates" img="https://mintcdn.com/claude-ai/F2QP9ubtYgT0edFI/images/claude-science/example-gallery/eng-beam-card-light.webp?fit=max&auto=format&n=F2QP9ubtYgT0edFI&q=85&s=ad60a68e22b34e917329cef00a950491" href="/docs/claude-science/example-gallery/eng-beam" width="1200" height="800" data-path="images/claude-science/example-gallery/eng-beam-card-light.webp" />

    <Card title="A first X-ray diffraction pattern: why silicon shows 6 peaks and salt 9" img="https://mintcdn.com/claude-ai/F2QP9ubtYgT0edFI/images/claude-science/example-gallery/mat-xrd-card-light.webp?fit=max&auto=format&n=F2QP9ubtYgT0edFI&q=85&s=6a52dca4ef4f80d0fabfbac9cec5e9d5" href="/docs/claude-science/example-gallery/mat-xrd" width="1200" height="800" data-path="images/claude-science/example-gallery/mat-xrd-card-light.webp" />

    <Card title="The Ulam spiral: are the primes' diagonal streaks real?" img="https://mintcdn.com/claude-ai/F2QP9ubtYgT0edFI/images/claude-science/example-gallery/math-ulam-card-light.webp?fit=max&auto=format&n=F2QP9ubtYgT0edFI&q=85&s=9fc25153a8d1957988c525fb51fd3f1e" href="/docs/claude-science/example-gallery/math-ulam" width="1200" height="800" data-path="images/claude-science/example-gallery/math-ulam-card-light.webp" />

    <Card title="The double slit, one electron at a time" img="https://mintcdn.com/claude-ai/F2QP9ubtYgT0edFI/images/claude-science/example-gallery/phys-doubleslit-card-light.webp?fit=max&auto=format&n=F2QP9ubtYgT0edFI&q=85&s=31cad5f730b79769fbe7f92147230b08" href="/docs/claude-science/example-gallery/phys-doubleslit" width="1200" height="800" data-path="images/claude-science/example-gallery/phys-doubleslit-card-light.webp" />

    <Card title="Schelling's segregation model: ask for 30% same-color neighbors, end up with 75%" img="https://mintcdn.com/claude-ai/F2QP9ubtYgT0edFI/images/claude-science/example-gallery/soc-schelling-small-card-light.webp?fit=max&auto=format&n=F2QP9ubtYgT0edFI&q=85&s=af39beeef336f122daf3fe993b885b1b" href="/docs/claude-science/example-gallery/soc-schelling-small" width="1200" height="800" data-path="images/claude-science/example-gallery/soc-schelling-small-card-light.webp" />
  </CardGroup>
</div>

claude-science/example-gallery/bio-drift New page · 35 lines, new page

# Genetic drift in a bottle: Buri's 107 fruit-fly populations ## Prompt ## Reference analysis ## Data

A whole new page. There's nothing to diff it against, so here is what it says.

# Genetic drift in a bottle: Buri's 107 fruit-fly populations

> Simulate 107 bottles of 16 fruit flies drifting by chance alone and compare them with Peter Buri's real bottles (1956).

<div className="example-gallery-page">
  <Frame>
    <img className="block dark:hidden" src="https://mintcdn.com/claude-ai/F2QP9ubtYgT0edFI/images/claude-science/example-gallery/bio-drift-full-light.webp?fit=max&auto=format&n=F2QP9ubtYgT0edFI&q=85&s=2e5d4d7b916e451a8f4d3fcdbcdb7dd9" alt="Three panels: 107 simulated allele-frequency paths over 19 generations, histograms at generations 1, 5, 10, and 19, and 20,000 replays at 16 and 9 breeding flies against Buri's 58." width="1600" height="706" data-path="images/claude-science/example-gallery/bio-drift-full-light.webp" />

    <img className="hidden dark:block" src="https://mintcdn.com/claude-ai/F2QP9ubtYgT0edFI/images/claude-science/example-gallery/bio-drift-full-dark.webp?fit=max&auto=format&n=F2QP9ubtYgT0edFI&q=85&s=781269cad496388cf0a1e54ab0d7af64" alt="Three panels: 107 simulated allele-frequency paths over 19 generations, histograms at generations 1, 5, 10, and 19, and 20,000 replays at 16 and 9 breeding flies against Buri's 58." width="1600" height="706" data-path="images/claude-science/example-gallery/bio-drift-full-dark.webp" />
  </Frame>
</div>

The numbers in your session might differ from this picture and from the summary in [Reference analysis](#reference-analysis).

## Prompt

Copy the prompt and paste it into a new session in [Claude Science](/docs/claude-science/overview). By default, Claude asks with a [permission card](/docs/claude-science/core-concepts#permission-cards) before it runs code or installs packages.

```text Prompt wrap theme={null}
In a 1956 paper, Peter Buri described keeping 107 bottles of fruit flies for 19 generations, restarting each bottle every generation with 8 males and 8 females picked at random. Every founder carried one copy each of two neutral eye-color alleles, bw and bw75, so bw75 started at a frequency of 0.5 in every bottle.

Simulate this with random sampling only: no selection, no mutation. Plot the bw75 frequency in all 107 populations over time, and draw histograms of the 107 frequencies at generations 1, 5, 10 and 19. Under each, print how many bottles are at 0, below 1/3, between 1/3 and 2/3, above 2/3, and at 1, and describe the change using only these 5 counts.

By generation 19, Buri found bw75 lost in 30 bottles and fixed in 28. Repeat your simulated experiment 1,000 times: how often does chance alone give that many? What population size would match his result? Compare it with Buri's own estimate for these bottles, about 9 breeding flies (2Ne = 17.86). Buri wrote that the causes are not known, but that unequal numbers of offspring per fly, down to sterility, are certainly part of it. This simulation does not test that, so report it as his view, in 1 or 2 sentences, and add no other causes. Explain each step as you would to an undergraduate.
```

## Reference analysis

107 simulated bottles of 16 fruit flies drift by chance alone: on average 25 lose or fix an allele in 19 generations (27 in the run pictured), but in Peter Buri's real bottles (1956) the count reached 58. A breeding population of about 9 gives 57, in line with Buri's own estimate (2Ne = 17.86): the textbook lesson of effective population size. Along the way the 107 frequencies, counted at generations 1, 5, 10, and 19, drain from the middle third into the outer thirds and from there into the 2 ends, 0 and 1, which no bottle leaves.

## Data

This example doesn't download any data.

Reference: Buri P., 'Gene frequency in small populations of mutant Drosophila', Evolution 10:367–402 (1956)

claude-science/example-gallery/chem-titration New page · 31 lines, new page

# A titration curve from the full equilibrium: which indicator finds the equivalence point? ## Prompt ## Reference analysis ## Data

A whole new page. There's nothing to diff it against, so here is what it says.

# A titration curve from the full equilibrium: which indicator finds the equivalence point?

> Solve the full equilibrium for 25.0 mL of 0.100 M acetic acid as 0.100 M NaOH is added, without the usual shortcuts.

<div className="example-gallery-page">
  <Frame>
    <img className="block dark:hidden" src="https://mintcdn.com/claude-ai/F2QP9ubtYgT0edFI/images/claude-science/example-gallery/chem-titration-full-light.webp?fit=max&auto=format&n=F2QP9ubtYgT0edFI&q=85&s=37e8dfb5f136ac8d3872d9cd328d04be" alt="Left: acetic acid's titration curve, with pH 2.88 at the start, 4.76 half-way, and 8.73 at equivalence. Right: acetic acid and HCl with the methyl orange and phenolphthalein ranges shaded: for acetic acid, methyl orange turns 17.4 mL too early." width="1425" height="674" data-path="images/claude-science/example-gallery/chem-titration-full-light.webp" />

    <img className="hidden dark:block" src="https://mintcdn.com/claude-ai/F2QP9ubtYgT0edFI/images/claude-science/example-gallery/chem-titration-full-dark.webp?fit=max&auto=format&n=F2QP9ubtYgT0edFI&q=85&s=34afada344f2f0f525f87785bab10ded" alt="Left: acetic acid's titration curve, with pH 2.88 at the start, 4.76 half-way, and 8.73 at equivalence. Right: acetic acid and HCl with the methyl orange and phenolphthalein ranges shaded: for acetic acid, methyl orange turns 17.4 mL too early." width="1425" height="674" data-path="images/claude-science/example-gallery/chem-titration-full-dark.webp" />
  </Frame>
</div>

The numbers in your session might differ from this picture and from the summary in [Reference analysis](#reference-analysis).

## Prompt

Copy the prompt and paste it into a new session in [Claude Science](/docs/claude-science/overview). By default, Claude asks with a [permission card](/docs/claude-science/core-concepts#permission-cards) before it runs code or installs packages.

```text Prompt wrap theme={null}
I'm titrating 25.0 mL of 0.100 M acetic acid (pKa 4.76) with 0.100 M NaOH. Compute the pH after each addition of base, from 0 to 50 mL, by solving the full equilibrium (mass balance, charge balance, Kw = 1.0 × 10⁻¹⁴) without the usual shortcuts (no Henderson–Hasselbalch, no 'x is small'), using concentrations, at 25 °C. Plot the titration curve and mark the starting point, the half-equivalence point and the equivalence point. Show that the acid alone sets the start, pH = pKa half-way, and acetate acting as a weak base sets the equivalence point.

Add the curve for 0.100 M HCl, and shade the color-change ranges of methyl orange (pH 3.1–4.4) and phenolphthalein (pH 8.3–10.0). Give the pH of both curves at 24.95 and 25.05 mL to show how steep each jump is. Which indicator works for which acid? If I picked the wrong one, how far off would my endpoint be, in mL (endpoint = completed color change)? Explain each step as you would to an undergraduate.
```

## Reference analysis

With acetic acid, phenolphthalein turns within 0.05 mL of the 25.00 mL equivalence point; methyl orange has fully turned at 7.6 mL, 17.4 mL too early. Methyl orange works for HCl (0.02 mL early). The reason: from 24.95 to 25.05 mL the pH runs from 4.0 to 10.0 for HCl but from 7.5 to 10.0 for acetic acid, entirely above methyl orange's range. Solving the full equilibrium for 25.0 mL of 0.100 M acetic acid as 0.100 M NaOH is added, without the usual shortcuts, using concentrations, at 25 °C, gives pH 2.88 at the start, 4.76 = pKa at the half-way point, and 8.73 at the equivalence point.

## Data

This example doesn't download any data.

claude-science/example-gallery/clin-km New page · 39 lines, new page

# A first Kaplan–Meier curve: the 1963 leukemia trial of 6-MP ## Prompt ## Reference analysis ## Data

A whole new page. There's nothing to diff it against, so here is what it says.

# A first Kaplan–Meier curve: the 1963 leukemia trial of 6-MP

> Compare 6-MP against placebo with Kaplan–Meier curves from 42 remission times typed into the prompt.

<div className="example-gallery-page">
  <Frame>
    <img className="block dark:hidden" src="https://mintcdn.com/claude-ai/F2QP9ubtYgT0edFI/images/claude-science/example-gallery/clin-km-full-light.webp?fit=max&auto=format&n=F2QP9ubtYgT0edFI&q=85&s=c4d20dad5701ebceb1d87c476424d2d3" alt="Two Kaplan–Meier plots: 6-MP and placebo remission curves with medians of 23 and 8 weeks (log-rank p = 0.00004), and the 6-MP curve three ways, with medians of 23, 16, and 10 weeks." width="1600" height="730" data-path="images/claude-science/example-gallery/clin-km-full-light.webp" />

    <img className="hidden dark:block" src="https://mintcdn.com/claude-ai/F2QP9ubtYgT0edFI/images/claude-science/example-gallery/clin-km-full-dark.webp?fit=max&auto=format&n=F2QP9ubtYgT0edFI&q=85&s=ecbd48885ccc0e663a9a65e93355234f" alt="Two Kaplan–Meier plots: 6-MP and placebo remission curves with medians of 23 and 8 weeks (log-rank p = 0.00004), and the 6-MP curve three ways, with medians of 23, 16, and 10 weeks." width="1600" height="730" data-path="images/claude-science/example-gallery/clin-km-full-dark.webp" />
  </Frame>
</div>

The numbers in your session might differ from this picture and from the summary in [Reference analysis](#reference-analysis).

## Prompt

Copy the prompt and paste it into a new session in [Claude Science](/docs/claude-science/overview). By default, Claude asks with a [permission card](/docs/claude-science/core-concepts#permission-cards) before it runs code or installs packages.

```text Prompt wrap theme={null}
In 1963 Freireich and colleagues published an early randomized, placebo-controlled trial of a chemotherapy drug: 42 patients under 20 with acute leukemia in remission got either 6-mercaptopurine (6-MP) or a placebo, and were followed until they relapsed or the trial was stopped, which happened early, once 6-MP was clearly ahead. These are the weeks each patient stayed in remission. A + means the patient was still in remission when the trial was stopped.

6-MP: 6, 6, 6, 6+, 7, 9+, 10, 10+, 11+, 13, 16, 17+, 19+, 20+, 22, 23, 25+, 32+, 32+, 34+, 35+
Placebo: 1, 1, 2, 2, 3, 4, 4, 5, 5, 8, 8, 8, 8, 11, 11, 12, 12, 15, 17, 22, 23

Draw the Kaplan–Meier curve of each arm, with tick marks for the + patients and the numbers at risk under the plot. Give the median remission time in each arm and compare the arms with a log-rank test. Then show me what happens to the 6-MP median if I count every + as a relapse, or if I leave those patients out. Explain each step as you would to a medical student.
```

## Reference analysis

Median remission 23 weeks on 6-MP against 8 on placebo (log-rank p = 0.00004): Kaplan–Meier curves from 42 remission times typed into the prompt. Mishandling the 12 patients still in remission gives a 6-MP median of 16 or 10 weeks instead. The data are the 42 patients analyzed when the trial Freireich and colleagues published in 1963 was stopped; both medians and χ² = 16.79 match the textbook values.

## Data

This example doesn't download any data.

References:

* Freireich E.J. et al., 'The effect of 6-mercaptopurine on the duration of steroid-induced remissions in acute leukemia: a model for evaluation of other potentially useful therapy', Blood 21(6):699–716 (1963), doi:10.1182/blood.V21.6.699.699
* Kleinbaum D.G. & Klein M., Survival Analysis: A Self-Learning Text, 3rd ed.

claude-science/example-gallery/cs-pagerank-tiny New page · 39 lines, new page

# PageRank on a web of 11 pages: why 1 good link beats 6 weak ones ## Prompt ## Reference analysis ## Data

A whole new page. There's nothing to diff it against, so here is what it says.

# PageRank on a web of 11 pages: why 1 good link beats 6 weak ones

> Follow a random surfer on Wikipedia's 11-page example until the numbers settle, and reproduce all 11 published values to 0.1 point.

<div className="example-gallery-page">
  <Frame>
    <img className="block dark:hidden" src="https://mintcdn.com/claude-ai/F2QP9ubtYgT0edFI/images/claude-science/example-gallery/cs-pagerank-tiny-full-light.webp?fit=max&auto=format&n=F2QP9ubtYgT0edFI&q=85&s=44bf56115d462e279420a8cfb04b1499" alt="Three panels: the 11-page web as circles sized by PageRank, the 11 shares settling to B 38.4%, C 34.3%, and E 8.1%, and bars showing that C's one link brings 32.7 points while E's six bring 6.5." width="1600" height="594" data-path="images/claude-science/example-gallery/cs-pagerank-tiny-full-light.webp" />

    <img className="hidden dark:block" src="https://mintcdn.com/claude-ai/F2QP9ubtYgT0edFI/images/claude-science/example-gallery/cs-pagerank-tiny-full-dark.webp?fit=max&auto=format&n=F2QP9ubtYgT0edFI&q=85&s=e857b5662f0295df70aaea333313232b" alt="Three panels: the 11-page web as circles sized by PageRank, the 11 shares settling to B 38.4%, C 34.3%, and E 8.1%, and bars showing that C's one link brings 32.7 points while E's six bring 6.5." width="1600" height="594" data-path="images/claude-science/example-gallery/cs-pagerank-tiny-full-dark.webp" />
  </Frame>
</div>

The numbers in your session might differ from this picture and from the summary in [Reference analysis](#reference-analysis).

## Prompt

Copy the prompt and paste it into a new session in [Claude Science](/docs/claude-science/overview). By default, Claude asks with a [permission card](/docs/claude-science/core-concepts#permission-cards) before it runs code or installs packages.

```text Prompt wrap theme={null}
PageRank is the idea that launched Google (Brin & Page, 1998): a page matters if pages that matter link to it. Here is a tiny web of 11 pages, A to K, where B→C means that page B links to page C:

B→C, C→B, D→A, D→B, E→B, E→D, E→F, F→B, F→E, G→B, G→E, H→B, H→E, I→B, I→E, J→E, K→E

Imagine a surfer who follows a random link on the current page 85 % of the time and jumps to a random page the rest of the time. From a page with no links, the surfer always jumps to a random page. Start with the surfer equally likely to be on any page, and repeat that step until the share of time spent on each page stops changing. That share is the page's PageRank.

Draw the web as circles with area proportional to PageRank, each letter inside its circle, C and E in color, no other labels, and plot how the 11 numbers settle. This is the example drawn in Wikipedia's PageRank article; its values are B 38.4 %, C 34.3 %, E 8.1 %, D and F 3.9 %, A 3.3 %, G to K 1.6 %. Then compare each page's PageRank with its share of the 17 links. C has 1 incoming link, but it is B's only link, and ranks 2nd; E has 6, all from low-ranked pages, and gets 8.1 %. Show how many points each link brings to C and to E. Explain each step as you would to an undergraduate.
```

## Reference analysis

Page C has 1 incoming link and ranks 2nd (34.3%); page E has 6 and gets 8.1%. Follows a random surfer on Wikipedia's 11-page example until the numbers settle, and reproduces all 11 published values to 0.1 point (B 38.4%, C 34.3%, E 8.1%, …). C's link is B's only link and brings 32.7 points; E's 6 links come from low-ranked pages and bring 6.5 points together; every page also gets the same 1.6-point floor. Draws the web as circles with area proportional to PageRank and shows the 11 numbers settling: every page is within 0.1 point of its final value after 29 steps.

## Data

This example doesn't download any data.

Example web after [PageRanks-Example.svg](https://commons.wikimedia.org/wiki/File:PageRanks-Example.svg) by 345Kai and Stannered (Wikimedia Commons, public domain).

Reference: S. Brin & L. Page, 'The anatomy of a large-scale hypertextual Web search engine', Computer Networks and ISDN Systems 30, 107–117 (1998)

claude-science/example-gallery/eng-beam New page · 29 lines, new page

# A first finite-element model: a clamped steel ruler bends and vibrates ## Prompt ## Reference analysis ## Data

A whole new page. There's nothing to diff it against, so here is what it says.

# A first finite-element model: a clamped steel ruler bends and vibrates

> Split a 300 mm steel ruler clamped at one end into 4 beam elements and check its bending under 1 N and its first 3 vibration modes against beam theory.

<div className="example-gallery-page">
  <Frame>
    <img className="block dark:hidden" src="https://mintcdn.com/claude-ai/F2QP9ubtYgT0edFI/images/claude-science/example-gallery/eng-beam-full-light.webp?fit=max&auto=format&n=F2QP9ubtYgT0edFI&q=85&s=f7c1012ed8af806d3ec445c6973cf630" alt="Three plots: the ruler bent under 1 N, tip 0.800 mm down as theory says; its first 3 mode shapes at 27.18, 170.53, and 480.62 Hz over theory; and frequency errors falling from 2 to 16 elements." width="1600" height="883" data-path="images/claude-science/example-gallery/eng-beam-full-light.webp" />

    <img className="hidden dark:block" src="https://mintcdn.com/claude-ai/F2QP9ubtYgT0edFI/images/claude-science/example-gallery/eng-beam-full-dark.webp?fit=max&auto=format&n=F2QP9ubtYgT0edFI&q=85&s=e0296e4f0767e1c3eaa160a9a62e67d4" alt="Three plots: the ruler bent under 1 N, tip 0.800 mm down as theory says; its first 3 mode shapes at 27.18, 170.53, and 480.62 Hz over theory; and frequency errors falling from 2 to 16 elements." width="1600" height="883" data-path="images/claude-science/example-gallery/eng-beam-full-dark.webp" />
  </Frame>
</div>

The numbers in your session might differ from this picture and from the summary in [Reference analysis](#reference-analysis).

## Prompt

Copy the prompt and paste it into a new session in [Claude Science](/docs/claude-science/overview). By default, Claude asks with a [permission card](/docs/claude-science/core-concepts#permission-cards) before it runs code or installs packages.

```text Prompt wrap theme={null}
I'm learning the finite-element method. Model a steel ruler clamped at one end: length 300 mm, width 25 mm, thickness 3 mm, E = 200 GPa, density 7850 kg/m³. Split it into 4 standard 2-node beam elements (cubic shape functions, consistent mass matrix) and write the matrices yourself with numpy. First hang 1 N (about 100 g) on the free end: beam theory says the tip drops PL³/(3EI) = 0.800 mm. Check that you get it, and draw the bent ruler over the theory curve. Then let the ruler vibrate: plot its first 3 mode shapes, each labeled with its frequency. Theory gives 27.18, 170.33 and 476.93 Hz; finite-element frequencies land slightly above and get closer with more elements, so show that in a small table for 2, 4, 8 and 16 elements. Give each frequency to 2 decimals, and its error in % worked out from the unrounded frequencies. Explain each step as you would to an undergraduate.
```

## Reference analysis

A 300 mm steel ruler clamped at one end is cut into 4 beam elements: under 1 N its tip drops 0.800 mm, exactly as beam theory says. Its first 3 vibration modes come out at 27.18, 170.53, and 480.62 Hz against 27.18, 170.33, and 476.93 Hz in theory, and 16 elements bring all 3 within 0.004%.

## Data

This example doesn't download any data.

claude-science/example-gallery/mat-xrd New page · 29 lines, new page

# A first X-ray diffraction pattern: why silicon shows 6 peaks and salt 9 ## Prompt ## Reference analysis ## Data

A whole new page. There's nothing to diff it against, so here is what it says.

# A first X-ray diffraction pattern: why silicon shows 6 peaks and salt 9

> Simulate the X-ray powder patterns of silicon and table salt and label every peak with Bragg's law.

<div className="example-gallery-page">
  <Frame>
    <img className="block dark:hidden" src="https://mintcdn.com/claude-ai/F2QP9ubtYgT0edFI/images/claude-science/example-gallery/mat-xrd-full-light.webp?fit=max&auto=format&n=F2QP9ubtYgT0edFI&q=85&s=6d56de735b8fe6c1fd456f7259fc7d4d" alt="Simulated X-ray diffraction patterns: silicon has 6 labeled peaks, with 200, 222, and 420 marked absent; rock salt has 9." width="1600" height="980" data-path="images/claude-science/example-gallery/mat-xrd-full-light.webp" />

    <img className="hidden dark:block" src="https://mintcdn.com/claude-ai/F2QP9ubtYgT0edFI/images/claude-science/example-gallery/mat-xrd-full-dark.webp?fit=max&auto=format&n=F2QP9ubtYgT0edFI&q=85&s=ce7e07ffd94f457100ac376aa7a18fab" alt="Simulated X-ray diffraction patterns: silicon has 6 labeled peaks, with 200, 222, and 420 marked absent; rock salt has 9." width="1600" height="980" data-path="images/claude-science/example-gallery/mat-xrd-full-dark.webp" />
  </Frame>
</div>

The numbers in your session might differ from this picture and from the summary in [Reference analysis](#reference-analysis).

## Prompt

Copy the prompt and paste it into a new session in [Claude Science](/docs/claude-science/overview). By default, Claude asks with a [permission card](/docs/claude-science/core-concepts#permission-cards) before it runs code or installs packages.

```text Prompt wrap theme={null}
X-ray diffraction is how we know where the atoms sit in a crystal. Simulate the powder diffraction patterns of silicon (diamond structure, a = 5.4311 Å) and rock salt (NaCl, a = 5.6402 Å) with pymatgen, for copper Kα₁ X-rays (wavelength="CuKa1", λ = 1.54056 Å) and angles 2θ up to 90°. Nothing needs downloading. Label every peak with its Miller indices hkl, and check each angle by hand with Bragg's law, λ = 2d sin θ, where d = a/√(h²+k²+l²). Both crystals share a face-centered cubic lattice, which allows 9 reflections in this range, yet silicon shows only 6: show that its 2-atom basis cancels 200, 222 and 420. In salt the all-odd peaks (111, 311, 331) are weak because Na and Cl scatter out of phase: show that too. Leave out other salts and what-ifs such as Na and Cl scattering equally, and round every number once, from its full-precision value. Put both patterns in 1 figure. Explain each step as you would to an undergraduate.
```

## Reference analysis

Simulates the X-ray powder patterns of silicon and table salt and labels every peak with Bragg's law. Both share a face-centered cubic lattice, yet below 2θ = 90° salt shows 9 peaks and silicon 6, because silicon's 2-atom basis cancels the 200, 222, and 420 reflections.

## Data

This example doesn't download any data.

claude-science/example-gallery/math-ulam New page · 35 lines, new page

# The Ulam spiral: are the primes' diagonal streaks real? ## Prompt ## Reference analysis ## Data

A whole new page. There's nothing to diff it against, so here is what it says.

# The Ulam spiral: are the primes' diagonal streaks real?

> Compare the share of primes per diagonal on Ulam's 1963 spiral of 1 to 40,401 against as many random odd numbers.

<div className="example-gallery-page">
  <Frame>
    <img className="block dark:hidden" src="https://mintcdn.com/claude-ai/F2QP9ubtYgT0edFI/images/claude-science/example-gallery/math-ulam-full-light.webp?fit=max&auto=format&n=F2QP9ubtYgT0edFI&q=85&s=140ca573e05d853818df9ab149c1f064" alt="Primes up to 40,401 on Ulam's spiral, with primes of n² + n + 41 in orange along 2 rays; as many random odd numbers, with no streaks; and histograms of each diagonal's share, wider for the primes." width="1600" height="632" data-path="images/claude-science/example-gallery/math-ulam-full-light.webp" />

    <img className="hidden dark:block" src="https://mintcdn.com/claude-ai/F2QP9ubtYgT0edFI/images/claude-science/example-gallery/math-ulam-full-dark.webp?fit=max&auto=format&n=F2QP9ubtYgT0edFI&q=85&s=fac8550c8919a1486851d6287563661b" alt="Primes up to 40,401 on Ulam's spiral, with primes of n² + n + 41 in orange along 2 rays; as many random odd numbers, with no streaks; and histograms of each diagonal's share, wider for the primes." width="1600" height="632" data-path="images/claude-science/example-gallery/math-ulam-full-dark.webp" />
  </Frame>
</div>

The numbers in your session might differ from this picture and from the summary in [Reference analysis](#reference-analysis).

## Prompt

Copy the prompt and paste it into a new session in [Claude Science](/docs/claude-science/overview). By default, Claude asks with a [permission card](/docs/claude-science/core-concepts#permission-cards) before it runs code or installs packages.

```text Prompt wrap theme={null}
In 1963 the mathematician Stanisław Ulam, doodling during a long and boring talk, wrote the whole numbers in a square spiral and circled the primes. Recreate his picture. Find all primes up to 40,401 (= 201 × 201) with a sieve of Eratosthenes, and check that 1,229 of them are below 10,000. Write 1, 2, 3, … in a square spiral starting from the center, and mark the primes.

Are the diagonal streaks real, or is my eye fooling me? Draw the same picture with the same number of randomly chosen odd numbers. For every diagonal of odd numbers at least 100 cells long, in both directions, compute the share of marked cells, and compare its standard deviation and range in the 2 pictures.

Then highlight the numbers n² + n + 41 (Euler's polynomial). On this spiral about the first 40 values wind around the center, and from then on they run along 2 straight diagonals. How many of the first 100 are prime, and how many would you expect from typical odd numbers of that size? The first one that is not prime is n = 40, which gives 41 × 41. Which diagonals are rich in primes is predicted by a conjecture of Hardy and Littlewood (1923) that is still unproven: mention it, and describe the rich and poor diagonals without accounting for them. Walk me through your method at undergraduate level.
```

## Reference analysis

Yes. On Ulam's 1963 spiral of 1 to 40,401, the share of primes per diagonal runs from 0 to 48%, against roughly 11 to 32% for as many random odd numbers. Over the 202 diagonals of at least 100 cells that spread is about 2.3 times the random one, so the streaks are real. The sieve finds 4,236 primes, 1,229 of them below 10,000, as published. Euler's n² + n + 41 traces the 2 richest streaks (over 70% prime along its 2 rays, each one half of a diagonal, running outward from the diagonal's cell nearest the center): prime for n = 0 to 39, and for 86 of its first 100 values, where typical odd numbers of that size would give about 29 of 100.

## Data

This example doesn't download any data.

Reference: M. L. Stein, S. M. Ulam and M. B. Wells, 'A visual display of some properties of the distribution of primes', American Mathematical Monthly 71, 516–520 (1964)

claude-science/example-gallery/phys-doubleslit New page · 35 lines, new page

# The double slit, one electron at a time ## Prompt ## Reference analysis ## Data

A whole new page. There's nothing to diff it against, so here is what it says.

# The double slit, one electron at a time

> Rebuild the picture of Tonomura's 1989 experiment with an ideal double slit: each electron lands as 1 random dot, yet fringes 10.7 µm apart build up.

<div className="example-gallery-page">
  <Frame>
    <img className="block dark:hidden" src="https://mintcdn.com/claude-ai/F2QP9ubtYgT0edFI/images/claude-science/example-gallery/phys-doubleslit-full-light.webp?fit=max&auto=format&n=F2QP9ubtYgT0edFI&q=85&s=d33485dbb10e2b0e60ce7127967ae21c" alt="Dots after 10 to 100,000 electrons build up into fringes; a histogram fit gives a spacing of 10.706 ± 0.003 µm against λL/d = 10.711 µm, and a smooth curve shows no fringes when the slit is known." width="1600" height="650" data-path="images/claude-science/example-gallery/phys-doubleslit-full-light.webp" />

    <img className="hidden dark:block" src="https://mintcdn.com/claude-ai/F2QP9ubtYgT0edFI/images/claude-science/example-gallery/phys-doubleslit-full-dark.webp?fit=max&auto=format&n=F2QP9ubtYgT0edFI&q=85&s=d56e5497b808e91e8b9c5bc59abf1650" alt="Dots after 10 to 100,000 electrons build up into fringes; a histogram fit gives a spacing of 10.706 ± 0.003 µm against λL/d = 10.711 µm, and a smooth curve shows no fringes when the slit is known." width="1600" height="650" data-path="images/claude-science/example-gallery/phys-doubleslit-full-dark.webp" />
  </Frame>
</div>

The numbers in your session might differ from this picture and from the summary in [Reference analysis](#reference-analysis).

## Prompt

Copy the prompt and paste it into a new session in [Claude Science](/docs/claude-science/overview). By default, Claude asks with a [permission card](/docs/claude-science/core-concepts#permission-cards) before it runs code or installs packages.

```text Prompt wrap theme={null}
In 1989 Akira Tonomura's team at Hitachi sent electrons one at a time past an electron biprism, the electron version of a double slit, and filmed an interference pattern building up dot by dot. Recreate that with an ideal double slit. Say nothing else about the 1989 experiment.

Take electrons accelerated through 50 kV (work out their de Broglie wavelength, with relativity), 2 slits 0.1 µm wide and 0.5 µm apart, and a screen 1 m away (check that d²/λL ≈ 0.05, so the simple far-field formula holds). Compute the probability of landing at each position on the screen. Then draw random hits from it, one electron at a time, and show the screen from −60 to 60 µm after 10, 100, 1,000, 10,000 and 100,000 electrons, noting on the figure how many hits land outside.

Fit the final pattern to measure the fringe spacing, and give it with its error bar: how many error bars from λL/d? Then show that the fringes vanish if we know which slit each electron went through. Explain each step as you would to an undergraduate.
```

## Reference analysis

Each electron lands as 1 random dot, yet fringes 10.7 µm apart build up. Rebuilds the picture of Tonomura's 1989 experiment with an ideal double slit: 50 kV electrons (de Broglie wavelength 5.355 pm, 2.4% shorter than without relativity), 2 slits 0.1 µm wide and 0.5 µm apart, a screen at 1 m. 10 electrons look random, 100 hint at bands, by 1,000 the fringes are plain, and at 100,000 a fit gives 10.71 µm, good to about 0.003 µm, matching λL/d = 10.711 µm. If we know which slit each electron went through, the fringes vanish and only the broad single-slit envelope is left.

## Data

This example doesn't download any data.

Reference: A. Tonomura, J. Endo, T. Matsuda, T. Kawasaki & H. Ezawa, 'Demonstration of single-electron buildup of an interference pattern', American Journal of Physics 57, 117–120 (1989)

claude-science/example-gallery/soc-schelling-small New page · 33 lines, new page

# Schelling's segregation model: ask for 30% same-color neighbors, end up with 75% ## Prompt ## Reference analysis ## Data

A whole new page. There's nothing to diff it against, so here is what it says.

# Schelling's segregation model: ask for 30% same-color neighbors, end up with 75%

> Ask for 30% same-color neighbors, end up with about 75%: the textbook computer version of Thomas Schelling's 1971 idea.

<div className="example-gallery-page">
  <Frame>
    <img className="block dark:hidden" src="https://mintcdn.com/claude-ai/F2QP9ubtYgT0edFI/images/claude-science/example-gallery/soc-schelling-small-full-light.webp?fit=max&auto=format&n=F2QP9ubtYgT0edFI&q=85&s=9150fb06019e6cb83311d70c1493c396" alt="A random board of blue and orange households (49% of neighbors alike), the same board settled into one-color patches (75%), and a curve of the share alike at the end against the demand, 51.3% to 99.6%." width="1600" height="596" data-path="images/claude-science/example-gallery/soc-schelling-small-full-light.webp" />

    <img className="hidden dark:block" src="https://mintcdn.com/claude-ai/F2QP9ubtYgT0edFI/images/claude-science/example-gallery/soc-schelling-small-full-dark.webp?fit=max&auto=format&n=F2QP9ubtYgT0edFI&q=85&s=d93cb22a2729f2eb5fb4d690c0d82206" alt="A random board of blue and orange households (49% of neighbors alike), the same board settled into one-color patches (75%), and a curve of the share alike at the end against the demand, 51.3% to 99.6%." width="1600" height="596" data-path="images/claude-science/example-gallery/soc-schelling-small-full-dark.webp" />
  </Frame>
</div>

The numbers in your session might differ from this picture and from the summary in [Reference analysis](#reference-analysis).

## Prompt

Copy the prompt and paste it into a new session in [Claude Science](/docs/claude-science/overview). By default, Claude asks with a [permission card](/docs/claude-science/core-concepts#permission-cards) before it runs code or installs packages.

```text Prompt wrap theme={null}
In 1971 the economist Thomas Schelling showed, with chips on a checkerboard, that a city can segregate sharply even when nobody asks for more than an even mix. Recreate the textbook computer version, which segregates at even milder demands. Fill a 50 × 50 grid at random with 1,125 blue and 1,125 orange households, leaving 250 cells empty. A household is content if at least 30 % of its occupied neighboring cells (the 8 around it, fewer at the edges) hold its own color. With 8 occupied neighbors, that means at least 3 of its own color, so it accepts 5 of the other. Each discontent household moves to a random empty cell; repeat until everyone is content. Show the grid before and after and measure the average share of like-colored neighbors: it starts near 50 %. Where does it end? How many points is that above the random board, and how many above the 30 % they asked for?

Then replace the 30 % by 10 %, 20 %, up to 70 % and plot what households get against what they ask for. Judge each result against the random board's 50 %, not against the demand: 10 % and 20 % stay within 10 points of it, the largest jump (about 18 points) is from 20 % to 30 %, and separation is nearly total from 60 %. Allow 500 rounds. Title each figure with what it shows. Give changes between thresholds in percentage points rather than in words like "half" or "triple". Explain each step as you would to an undergraduate.
```

## Reference analysis

Ask for 30% same-color neighbors, end up with about 75%. On a 50 × 50 grid filled at random with 1,125 blue and 1,125 orange households and 250 empty cells (about 50% of neighbors alike), only about 18% of households are discontent. Yet once each discontent one has moved to a random empty cell (roughly 1,000 moves in 10 to 20 rounds), the board is a patchwork of 1-color neighborhoods. Redone for demands of 10% to 70%, 20 random boards each: 10% leaves the random board as it is (51%), 20% lifts it to 57%, 50% gives 88% (about 7 like neighbors for each unlike one), and separation is nearly total from 60% (98% at 60%, 99.6% at 70%). The textbook computer version of Thomas Schelling's 1971 idea.

## Data

This example doesn't download any data.

Reference: T. C. Schelling, 'Dynamic models of segregation', Journal of Mathematical Sociology 1(2), 143–186 (1971)

third-party/claude-desktop/configuration Changed · +5 / -5 lines