Hello, World!

Finite Geometry Explorer

Click ANY point in the finite matrix

Point Explorer

Click a point

Euclidean
√(x² + y²)
—

Taxicab
|x| + |y|
—

Square
max(|x|, |y|)
—

Circle: x² + y² ≤ 1

Taxicab: |x| + |y| ≤ 1

Square: |x| ≤ 1, |y| ≤ 1

Waiting for a point...

Bongiornotes — Discrete Geometry

BONGIORNOTES

Discrete Pixel Geometry Laboratory

1. Discrete Circle

Construct a circular region from square grid cells.

Radius: 5

10 Diameter (grid units)

— Filled cells

20 4 × radius

40 8 × radius

2. Spherical Grid

A rotating projection of latitude and longitude lines on a sphere.

90° Right angle

360° Full rotation

2R Diameter

4 × 90° Four right-angle sectors

3. The Discrete Geometry Framework

The framework begins with a square grid and treats the grid cells as discrete geometric units.

The center of the construction is a reference point. A radius R extends from that center to the boundary.

In the proposed arc convention, one quarter-turn is assigned a path length of 2R. Four such quarter-turns therefore give:

2R + 2R + 2R + 2R = 8R

Comparing that convention with the standard Euclidean circumference relationship

C = 2πR

produces the proposed relationship

2πR = 8R → π = 4

Important: π = 4 is not the value of π in standard Euclidean geometry. Here it is presented as the result of the framework's alternative discrete arc-length convention. The purpose of the experiment is to compare the two geometric models.

Hello, World!

Circle Geometry Explorer

Cosine & Sine → Coordinates → Circle

Euclidean Circle Taxicab Circle Active Vector

Angle θ

Radius r

EUCLIDEAN CIRCLE

x = r cos(θ)

y = r sin(θ)

x = 1 × cos(45°) = 0.7071
y = 1 × sin(45°) = 0.7071

TAXICAB CIRCLE

x = r cos(θ) / (|cos(θ)| + |sin(θ)|)

y = r sin(θ) / (|cos(θ)| + |sin(θ)|)

x = 0.7071 / 1.4142 = 0.5000
y = 0.7071 / 1.4142 = 0.5000

Euclidean Point

X: 0.7071

Y: 0.7071

Distance: 1.0000

Taxicab Point

X: 0.5000

Y: 0.5000

L₁ Distance: 1.0000

Euclidean
x² + y² = r²

Taxicab
|x| + |y| = r