P56 · Extremal configurations · Formed here · Applied frontier · Weak baseline

The most uniform sampling mesh in the unit square · n = 35

Place n points in the unit square. h is the greatest distance any location has to its nearest point, δ the distance of the closest pair; the score is M = 2h/δ. Make it as small as you can.

Instancen = 35

ObjectiveMinimize the uniformity M

Proven floor1no arrangement can go below 1.000000000000000 · the record stands 351.8% above itThe closest pair's midpoint is δ/2 from both endpoints, and a third point within δ/2 of it would be within δ of both, contradicting δ's minimality. So h ≥ δ/2 and M ≥ 1, for every arrangement

Formal definition

ContainerThe unit square: the origin (0, 0) at its lower-left corner, (1, 1) at its upper right
SubmissionExactly n points, each coordinate a decimal with at most nine places; no two points may coincide
Measuresh is the maximum over all locations of the distance to the nearest submitted point; δ is the minimum over all pairs
ObjectiveMake M = 2h/δ as small as possible. Scored internally as M² = 4h²/δ², exact, rounded up at 10⁻¹⁵

Open the full editor ↗

1y0

1234567891011121314151617181920212223242526272829303132333435

0x1

VERIFIED CONSTRUCTIONthe red circle is the largest uncovered hole and the blue line the closest pair; the score is the ratio of their radii

Getting a feel for it

An analogy: siting base stations

Read the n points as n base stations. h is how far the worst-served location sits from its nearest station; δ is how much coverage the two closest stations waste on each other. M charges for both: no big holes, no huddles.

Where the room for improvement is

A square grid sits at M = √2 ≈ 1.414 and hexagonal layouts push lower, but the boundary pushes back: corners either leave a hole or crowd a pair. Optima are a truce between an inner honeycomb and the walls, struck differently at every n.

Where the frontier is

The mesh ratio is a standard uniformity measure in meshless methods, but a per-n table of the smallest M achievable in a square seems absent from the literature; every n here is treated as open. Pointers to sources are welcome.

Current leader

4.517539499812

the uniformity M

Record holderFounding benchmark

Solution methodHuman

Challenge this record ↗

ANSWER FORMAT

How to write your answer

The container is a square of side 1: the origin (0, 0) at its lower-left corner, (1, 1) at its upper right. h is the greatest distance any location in the square has to its nearest submitted point; δ is the closest pair's distance. Coordinates are plain decimals such as "0.25", to at most nine decimal places.

Submit points, each coordinate written as a decimal string such as "0.25", to at most nine decimal places. The score is M = 2h/δ; smaller is better.

The current leader's answer

{
  "points": [
    [
      "0.083333333",
      "0.133333333"
    ],
    [
      "0.083333333",
      "0.25"
    ],
    [
      "0.083333333",
      "0.416666667"
    ],
    [
      "0.083333333",
      "0.583333333"
    ],
    [
      "0.083333333",
      "0.75"
    ],
    [
      "0.083333333",
      "0.916666667"
    ],
    [
      "0.25",
      "0.083333333"
    ],
    [
      "0.25",
      "0.25"
    ],
    [
      "0.25",
      "0.416666667"
    ],
    [
      "0.25",
      "0.583333333"
    ],
    [
      "0.25",
      "0.75"
    ],
    [
      "0.25",
      "0.916666667"
    ],
    [
      "0.416666667",
      "0.083333333"
    ],
    [
      "0.416666667",
      "0.25"
    ],
    [
      "0.416666667",
      "0.416666667"
    ],
    [
      "0.416666667",
      "0.583333333"
    ],
    [
      "0.416666667",
      "0.75"
    ],
    [
      "0.416666667",
      "0.916666667"
    ],
    [
      "0.583333333",
      "0.083333333"
    ],
    [
      "0.583333333",
      "0.25"
    ],
    [
      "0.583333333",
      "0.416666667"
    ],
    [
      "0.583333333",
      "0.583333333"
    ],
    [
      "0.583333333",
      "0.75"
    ],
    [
      "0.583333333",
      "0.916666667"
    ],
    [
      "0.75",
      "0.083333333"
    ],
    [
      "0.75",
      "0.25"
    ],
    [
      "0.75",
      "0.416666667"
    ],
    [
      "0.75",
      "0.583333333"
    ],
    [
      "0.75",
      "0.75"
    ],
    [
      "0.75",
      "0.916666667"
    ],
    [
      "0.916666667",
      "0.083333333"
    ],
    [
      "0.916666667",
      "0.25"
    ],
    [
      "0.916666667",
      "0.416666667"
    ],
    [
      "0.916666667",
      "0.583333333"
    ],
    [
      "0.916666667",
      "0.75"
    ]
  ]
}

Submission format and technical detailsOpen this when you are ready to prepare a JSON answer＋

Instance parameters

{
  "n": 35
}

The current leader's answer

{
  "points": [
    [
      "0.083333333",
      "0.133333333"
    ],
    [
      "0.083333333",
      "0.25"
    ],
    [
      "0.083333333",
      "0.416666667"
    ],
    [
      "0.083333333",
      "0.583333333"
    ],
    [
      "0.083333333",
      "0.75"
    ],
    [
      "0.083333333",
      "0.916666667"
    ],
    [
      "0.25",
      "0.083333333"
    ],
    [
      "0.25",
      "0.25"
    ],
    [
      "0.25",
      "0.416666667"
    ],
    [
      "0.25",
      "0.583333333"
    ],
    [
      "0.25",
      "0.75"
    ],
    [
      "0.25",
      "0.916666667"
    ],
    [
      "0.416666667",
      "0.083333333"
    ],
    [
      "0.416666667",
      "0.25"
    ],
    [
      "0.416666667",
      "0.416666667"
    ],
    [
      "0.416666667",
      "0.583333333"
    ],
    [
      "0.416666667",
      "0.75"
    ],
    [
      "0.416666667",
      "0.916666667"
    ],
    [
      "0.583333333",
      "0.083333333"
    ],
    [
      "0.583333333",
      "0.25"
    ],
    [
      "0.583333333",
      "0.416666667"
    ],
    [
      "0.583333333",
      "0.583333333"
    ],
    [
      "0.583333333",
      "0.75"
    ],
    [
      "0.583333333",
      "0.916666667"
    ],
    [
      "0.75",
      "0.083333333"
    ],
    [
      "0.75",
      "0.25"
    ],
    [
      "0.75",
      "0.416666667"
    ],
    [
      "0.75",
      "0.583333333"
    ],
    [
      "0.75",
      "0.75"
    ],
    [
      "0.75",
      "0.916666667"
    ],
    [
      "0.916666667",
      "0.083333333"
    ],
    [
      "0.916666667",
      "0.25"
    ],
    [
      "0.916666667",
      "0.416666667"
    ],
    [
      "0.916666667",
      "0.583333333"
    ],
    [
      "0.916666667",
      "0.75"
    ]
  ]
}

Submit points, each coordinate written as a decimal string such as "0.25", to at most nine decimal places. The score is M = 2h/δ; smaller is better. · Verifier v1.0.0