Escher-Style Fish Tile

By the end of this lab you'll be able to:

• Store a shape outline as a list of (x, y) waypoints
• Explain the tessellation rule: whatever bumps OUT of one edge must notch IN on the opposite edge
• Tile a shape across the screen by translation, alternating two colors

The artist M.C. Escher filled entire walls with lizards, birds, and fish that lock together with no gaps and no overlaps. His secret was simple and brilliant: start with a shape that tiles (like a rectangle), and every time you push a bump out of one edge, carve the exact same bump into the opposite edge. This program builds a fish that way — its nose is the bump, its tail is the matching notch — then stamps 16 copies in a perfect interlocking school.

Welcome to the Fish Factory!

Escher's fish look impossible to design — but they start as a plain rectangle! Bend the left edge into a nose, cut the same shape from the right edge as a tail, and the copies snap together like puzzle pieces.

How It Works

The fish outline is a list of waypoints tracing an 80×50 rectangle whose edges have been reshaped in matching pairs. The left-edge nose points (-14, 35), (-6, 25), (-14, 15) reappear on the right edge shifted by exactly +80: (66, 35), (74, 25), (66, 15). The bottom-edge belly dip at (40, -8) reappears on the top edge shifted by +50 as (40, 42). Because every bump matches a notch one tile-width away, copies placed every 80 pixels across and 50 pixels up fit with zero gaps.

Sample Code

import turtle
monty = turtle.Turtle()
monty.speed(0)
monty.hideturtle()

outline = [(0, 0), (25, 0), (40, -8), (55, 0), (80, 0),
           (66, 15), (74, 25), (66, 35), (80, 50),
           (55, 50), (40, 42), (25, 50), (0, 50),
           (-14, 35), (-6, 25), (-14, 15)]

def draw_fish(x, y, color):
    monty.penup()
    monty.color(color, color)
    monty.goto(x, y)
    monty.pendown()
    monty.begin_fill()
    for px, py in outline:
        monty.goto(x + px, y + py)
    monty.goto(x, y)
    monty.end_fill()
    monty.penup()
    monty.goto(x + 6, y + 31)
    monty.dot(7, 'black')
    monty.dot(3, 'white')

for row in range(4):
    for col in range(4):
        x = -150 + col * 80
        y = -100 + row * 50
        if (row + col) % 2 == 0:
            draw_fish(x, y, 'coral')
        else:
            draw_fish(x, y, 'lightseagreen')

What Do You Think Will Happen?

The color is picked by (row + col) % 2 — a checkerboard rule. Will the fish directly ABOVE a coral fish be coral or sea-green? Decide, then run the program and check one column.

Try It Now

▶ Run ↻ Reset

Sea-green — moving up one row changes row + col from even to odd (or odd to even), so the color always flips. Were you right?

How It Works

Check the interlocking for yourself with coordinates. The fish at column c draws its tail notch at absolute points c*80 + (66, 15), c*80 + (74, 25), c*80 + (66, 35). Its right-hand neighbor starts at (c+1)*80 and draws its nose at (c+1)*80 - 14 = c*80 + 66 and (c+1)*80 - 6 = c*80 + 74 — the identical points, traversed the other way. One fish's tail water is the next fish's nose. The same trick, shifted by 50 instead of 80, makes each fish's arched back cradle the belly of the fish above it.

Explanation Table

Line	What it does
outline = [(0, 0), (25, 0), (40, -8), ...]	The fish shape as waypoints around a modified rectangle
(-14, 35), (-6, 25), (-14, 15)	The nose bump — exactly matches the tail notch 80 px to the right
monty.goto(x + px, y + py)	Translates every waypoint to this fish's grid position
if (row + col) % 2 == 0:	Checkerboard rule that alternates the two fill colors

Learning Check

Find the Bug

Someone bent this fish's tail too deep, and now there are white diamond-shaped gaps in the school — Escher would not approve! Compare the tail waypoints with the nose waypoints. One number breaks the "bump equals notch shifted by 80" rule. Fix it!

▶ Run ↻ Reset

The tail's middle waypoint is (60, 25), but the nose's middle waypoint is (-6, 25), and -6 + 80 = 74. Change (60, 25) back to (74, 25) and the white diamonds vanish — bump and notch match again.

Experiments

1. New school colors. Change 'coral' to 'gold' and 'lightseagreen' to 'slateblue'. You'll know it worked when the checkerboard of fish switches to the new pair everywhere.

2. Longer nose. Change both nose points (-14, 35) and (-14, 15) to (-24, 35) and (-24, 15), AND the matching tail points (66, 35) and (66, 15) to (56, 35) and (56, 15). You'll know it worked when every fish gets a pointier face and the school still has zero gaps.

3. Bigger eye. Change monty.dot(7, 'black') to monty.dot(11, 'black') and the white dot from 3 to 5. You'll know it worked when all 16 fish look wide awake.

4. Deeper belly. Change the belly point (40, -8) to (40, -16) and the back point (40, 42) to (40, 34). You'll know it worked when the fish look plumper and each row still nests perfectly into the row above.

You Out-Tiled Escher!

Now you know the tessellation secret: every bump OUT must be a matching notch IN on the opposite edge. With that one rule you can turn rectangles into fish, birds, lizards — anything! Up next: Voronoi Color Regions — color the whole canvas by which seed point is closest.