# Can you find the center of gravity?

> Interactive mechanical perception experiment published at `https://mariuscomper.uk/punctul-de-echilibru/en/`.

## Premise

A rigid brass plate balances on a needle point at exactly one location: its physical center of gravity. Any offset creates a gravitational moment of torque that topples the plate.

## Physical Mechanism & Visual Bias

The human brain does not solve surface integrals when viewing an object. Instead, visual perception relies on two heuristic shortcuts:
1. **Bounding-box centering**: Seeking the midpoint of the outer framing rectangle.
2. **Mass-clustering pull**: Placing the needle toward the bulky lobe while underestimating the moment arm of thin arms.

According to Archimedes' law of the lever in *On the Equilibrium of Planes* (250 BC):
\[
\vec{\tau} = \iint (\vec{r} - \vec{r}_p) \times d\vec{F}_g = \vec{0}
\]
A single gram 100 millimeters from the pivot exerts the exact same overturning torque as ten grams resting 10 millimeters away.

## Mathematical Formulation (Green's Theorem)

For a uniform planar polygon with \(n\) vertices:
- **Area**: \(A = \frac{1}{2} \sum_{i=0}^{n-1} (x_i y_{i+1} - x_{i+1} y_i)\)
- **Centroid X**: \(C_x = \frac{1}{6A} \sum_{i=0}^{n-1} (x_i + x_{i+1}) (x_i y_{i+1} - x_{i+1} y_i)\)
- **Centroid Y**: \(C_y = \frac{1}{6A} \sum_{i=0}^{n-1} (y_i + y_{i+1}) (x_i y_{i+1} - x_{i+1} y_i)\)

## The 5 Specimens

1. **Tapered Wedge**: The heavy base concentrates mass; the needle tip draws the eye rightward.
2. **Asymmetric L-Bracket**: The visual bounding center falls in empty space; the true centroid sits near the interior corner.
3. **Swept Boomerang**: Curvature pulls mass toward the inner concave crook.
4. **Stepped Anvil**: Cantilevered horn shifts the framing midpoint away from primary mass.
5. **Faceted Arrowhead**: The wide base retains mass behind, while the elongated tip draws the eye forward.

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Marius Comper · https://mariuscomper.uk