Lecture 23 · Mathematics
Locate yourself from a tiny picture
Video in production
This lecture is still face down. Its script is written; filming and animation are under way.
Meanwhile, read the outlineWhat you will learn
Decode a position from a verified 2×2 binary patch while respecting toroidal boundaries, orientation and sensor shape.
Before this lecture
Ideas you will use
- Cyclic window codes
- Basic row/column coordinates
In this lecture
An outline of the lecture, scene by scene, with the statements that appear on screen. It follows the draft script, so details may change in production.
Pose local position question on grid
- 16 sensor origins
- 4 bits→16 possible patches
- North fixed; edges wrap
Present exact torus and read first window
- Rows 1101/0001/1000/1011
- Read top row then bottom row
- (row 1, col 1)→1100
Explain wrap edges and corners exactly
- Corner(4,4)→1111
- Wrap does not rotate the image
- Location is modulo 4 in each direction
Verify all 16 patches and build lookup
- 16 origins; 16 distinct patches
- 0000→(2,2)
- 1111→(4,4)
Show orientation as additional unknown not free
- Known orientation required
- 16×4 states >16 reports
- Rotated image can be valid elsewhere
Compare window shapes and noise
- Sensor shape is part of the code
- Horizontal 1000 collides
- No automatic bit-error detection
Companion localization game with failure modes
- Locate within one periodic cell
- Worked 0000→(2,2)
- Diagnose orientation, mask and bit errors
Connect applications carefully and retrieve
- Local observation→global address
- Conditions make the claim precise
- Next: reconstruction from fragments
Companion interactive
Planned
Locate each 2×2 patch within the 4×4 periodic cell.
The interactive for this lecture is being built and will appear here.
Notes
Written notes for this lecture will appear here.
Sources
- Magical Mathematics, pp. 30–34
How these sources were used
- Exact 4×4 two-dimensional de Bruijn pattern and localization motivation.