Twenty studies of what sameness can—and cannot—tell us. These figures use the actual constructions, on shared axes where comparisons require them.
Same count. Different claims.
spherical sampling
Count does not determine coverage, dependence, or suitability for a task. The spacing statistic answers one specific question.
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Same projection. Different bodies.
occupancy from a projection
A single projection hides occupied depth. Turn the body, then return to the canonical face to recover the invariant.
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Same planet. Different frames.
measure vs shape
The sphere is the reference surface. Its globe view is an orthographic projection too; Mercator and Equal Earth make different local trade-offs.
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Same generators. Different territory.
nearest-site territory
The generators stay put. Changing the distance rule changes which generator owns each location.
Enter live exhibit →Same samples. Different assumptions.
model / influence
Inspect the response to a hypothetical +1 at one sample. The reach of that change reveals the assumption behind each interpolant.
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Same target budget. Different obligations.
structure / allocation
One target budget serves three objectives. The achieved mean is numerical and shown explicitly; lower compliance means better only for its named objective.
Same x. Same y. Different pairing.
coupling
The same x values and the same y values permit different pairings. Marginal information does not determine joint behavior.
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Same averages. Different groups.
aggregation / weights
Keep all 400 records and both pooled rates. Change group membership to make the within-group difference agree, disappear, or reverse.
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Same Fourier magnitude. Different structure.
spectrum vs structure
Phase carries structure that a Fourier magnitude array cannot recover. The three images use one shared grayscale.
Enter live exhibit →Same rule. Different histories.
dynamics / prediction
A fixed update rule does not specify an initial state. Three nearby starts follow exactly the same rule into different histories.
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Same moves. Different place.
rigid motion / composition
Every ordering shares its total rotation and length, but the endpoint depends on composition order. Turn scale λ changes the moves; zero makes them commute.
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Same sources. Different flow.
vector fields / underdetermination
Equal divergence and witness flux do not determine the velocity field. Added divergence-free components change curl, circulation and trajectories.
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Same neighbors counted. Different worlds.
networks / reachability
Send a pulse from one vertex. Three exact cubic graphs keep every degree at three while their routes and connected components change.
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Same push integrated. Different motion.
forced linear dynamics
Equal applied-force area does not fix a spring–mass–damper’s motion. Time the second pulse and compare peak displacement and the residual vibration at one shared instant.
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Same eventual stability. Different excursions.
linear dynamics / transient growth
The eigenvalues stay negative as coupling changes. With one fixed start, coordinate system, and norm, the state can first travel farther from zero before it decays.
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Same distances. Different handedness.
distance geometry
A labeled tetrahedron and its mirror share all six edge lengths. Their opposite signed volumes forbid a proper rigid match: the certified best 3D RMS error is 1 unit, while reflection aligns them exactly.
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Same residual size. Different error.
linear algebra / conditioning
A unit residual rotates through one fixed linear system. The corresponding error traces a 100-to-1 ellipse; relative amplification varies from 1 to 100 while the condition-number bound stays fixed.
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Same fitted summary. Different structure.
statistics / regression diagnostics
Anscombe’s four datasets share rounded means, variances, correlation and fitted coefficients. Their actual computed fits differ slightly, while their point arrangements and residual patterns differ visibly.
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Same numbers added. Different totals.
floating point / rounding
Real addition forgets order; a binary64 accumulator does not. Each addition rounds to the nearest representable value, whose spacing depends on the result’s magnitude. An exact BigInt ledger records signed corrections, including upward rounding.
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Same forces at the supports. Different bend.
beam mechanics / equilibrium
Every arrangement has the same total load and first moment, so each support reads W/2. The load distribution changes the internal moment and deflection. At the quarter-point setting, the pair and uniform load even share a peak moment, yet their midspan deflections differ by 10%.
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