Lecture 17 · Mathematics
From a card window to a directed edge
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
Translate overlapping binary windows into edges of a directed graph and back without confusing vertices with edges.
Before this lecture
Ideas you will use
- Binary cycles
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.
Make overlap into a movement rule
- 001→011 shares 01
- 001→010 also legal
- Word tiles will become arrows
Define vertices as contexts and edges as words
- Vertex=2-bit context
- Edge=3-bit word
- 001: 00→01
Build all eight edges and distinguish repeated vertices
- 8 edges=8 words
- Each vertex: 2 in,2 out
- Repeat vertices; do not repeat edges
Read a particular closed walk into a cyclic word
- Start 00; append each edge’s last bit
- 0001110100→cyclic 00011101
- Closure supplies the wraparound
Explain reverse translation and its limitations
- Continuous: overlaps agree
- Edge-once: words are unique
- Closed: ends fit
Generalize to order k with explicit k 4 example
- Order k: 2ᵏ⁻¹ vertices,2ᵏ edges
- 0010: 001→010
- Append target bits to reach target context
Companion overlap puzzle with exact success condition
- Goal: all 8 edges once and return
- Partial tour≠complete Euler tour
- Word↔edge; overlap↔vertex
Retrieve correspondences and preview proof
- Vertex=context
- Edge=word
- Euler tour=every word once
Companion interactive
Planned
Build a continuous closed walk using all 8 word edges once.
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. 20–21
How these sources were used
- de Bruijn graph and Euler-tour correspondence.