LEARN BY CHANGING ONE THING
A few small experiments.
A way into the mathematics.
Move a slider. Remove an object. See what changes—and why. These examples connect simple pictures to the questions in our research.
Interactive teaching examples
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Current research sources
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01 / THE SHAPE NEAR A POINT
When is a hill really a peak?
At a stationary point (a critical point), the slope is zero. What matters next is how the surface bends. If it bends downward in every direction, the point is a local peak. A saddle bends upward in one direction and downward in another.
Try this: move the center curvature upward through −1, then through +1.
Moves both principal curvatures together.
Separates the two curvatures: s − R and s + R.
Choose a quadratic example, or adjust the sliders. At the singular boundary, the second-order test is inconclusive.
Sliders work with the arrow keys.
LOCAL SHAPE
Downward in both directions
First curvature−3.0
Second curvature−1.0
Both curvatures are negative: this quadratic model bends downward in every direction.
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Figure citation and export limits
The SVG contains the current diagram, concrete display colors, labels, a generation timestamp and recorded source identity. It uses local fonts; glyph shapes can vary between viewers. Its permalink includes only the two curvature settings.
Why this idea appears in the research
A Hessian is a matrix that records curvature. The coefficient calculation sorts matrices by their curvature signs, then averages a weighted contribution over the peak-shaped ones. The picture lets you inspect that selection rule.
See the mathematics, source and limits
The source writes the two-by-two transverse Hessian as A = [[s + x, y], [y, s − x]], with R = √(x² + y²). Its eigenvalues are s − R and s + R. Negative definiteness is exactly s < −R. At a zero eigenvalue, the second-order test is inconclusive; higher-order terms can matter.
This deterministic diagram is a slice through those parameters. It does not sample the source’s Gaussian matrix law or compute its average. Numeric rendering uses floating point and is non-certifying.
Read the pinned SIDE24 derivation, §1 ↗ · Inspect the exact coefficient endpoints
02 / TWO DIFFERENT KINDS OF SMALL
Closer together. Much closer in height.
Call the distance between two specified critical points r. In the research model, their height gap is proportional to r³. Halving the distance divides that gap by eight. Meanwhile, different regions of space can shrink—or stay fixed.
Try this: compare r = 0.50 with r = 0.25. Then switch the highlighted region.
Illustrative units; height gap = r³ in this example.
The comparison button switches between 0.50 and 0.25.
SPACE AND HEIGHT
The ring follows the points inward
Point separation0.50
Height gap r³0.125000
Remote boundary ρ (fixed)3.00
The ring spans radii 1.00 to 2.00. The remote boundary stays at 3.00.
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Approaching-points citation and export limits
The SVG retains the current spatial diagram and separate height scale, accessible description, display colors, generation time and recorded source identities. The small height window remains on the fixed scale. Its link restores only separation and region. Local font shapes can vary between viewers.
A small distinction with a big effect
The fixed-annulus argument concerns the shrinking ring (annulus) above and allows all heights. The fixed-remote argument keeps a spatial exclusion fixed and counts within the shrinking height window. Their conditions answer different questions.
See the mathematics, sources and limits
The diagram fixes A = 2, B = 4, ρ = 3, L = 24, b = 1 and k = 1 to illustrate the definitions. Pins are M = (−r/2, 0) and S = (r/2, 0). The annulus is Ar ≤ |x| ≤ Br; the remote region satisfies dist(x, 0) ≥ ρ; the remote height window is b − kr³ < f(x) < b.
These selected dimensions are teaching choices, not certified theorem constants or an admissible radius bound. The viewport is a local schematic, not the whole torus. This example displays scaling only; it estimates no critical-point count, probability or lifetime. Numeric rendering is non-certifying. The linked sources retain their own hypotheses and review histories.
Pinned fixed-remote source ↗ · Pinned fixed-annulus source ↗ · Read the source-bound exhibit
03 / UNDERSTANDING AN OBSTRUCTION
Three objects. Two places. One impossible fit.
Suppose you have two groups, and each group can hold at most one object. Can all three objects be assigned? What if you remove any one of them?
Try this: remove any one object. Put it back, then remove a different one.
The rule stays the same: one object per group.
Use Tab to reach an object and Space to select it.
CAPACITY ONE · TWO GROUPS
One object has no place
Three included objects need three places, but only two are available. Removing any one object makes a valid assignment possible.
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Finite-obstruction citation and export limits
The SVG records the included objects, the displayed assignment or failed attempt, accessible description, display colors, generation time and pinned source identity. Its link restores only the object selection. Local font shapes can vary between viewers.
The whole set can fail while every smaller set works
That makes the triple a minimal obstruction: it fails the rule, but removing anything repairs it. This tiny example is a starting point for the project’s work on finite capacity families — the full-price theorem, whose stated scope is set out on the Research page.
See the mathematics, source and limits
This reuses the museum’s finite P15 model: one original block, capacity a = 1, demand d = 2, n = 3 coordinates, K = 2 palette classes and no macro edges. A valid group is a member of the capacity family; a valid assignment partitions the selected objects into at most two such groups.
The full triple is not two-decomposable, and every proper subset is. This exact finite example illustrates one realized family. It does not establish the full price theorem, extend to arbitrary families, or close a prize problem.
Read the pinned full-price proof, §5 ↗ · Inspect its source-bound exhibit
KEEP GOING
Take the picture into the proof.
The research pages connect each mathematical statement with its exact scope, sources and review record.
Next step: turn synthetic counts into a density, a worked measurement example.
These teaching examples explain definitions. They do not change the status of any research claim.