Demonstration 1 of 4
Carry every link through the pairing
The square and the diamond look different. Does a pairing of their points keep every link?
Each link of the square is redrawn on the diamond using the pairing. A solid teal line lands on a listed link; a dashed line lands on a pair the diamond does not link.
\[A-B, A-C, B-D, C-D\]
\[P-Q, P-R, Q-S, R-S\]
A, B, C and D are the vertices of the first drawing; P, Q, R and S are the vertices of the second. A-B means a link joins A and B. An arrow such as A → P pairs a vertex with its partner. Carrying a link means replacing both of its endpoints.
Predict first. Try the pairing that sends B to R and C to Q. Will all four carried links still land on listed links?
Choose an example
Constructed example: the chapter's Figure 4.1 networks and its opening pairing, plus two other pairings.
Calculated values
- Pairing
- A → P, B → Q, C → R, D → S
- Link A-B
- P-Q (listed)
- Link A-C
- P-R (listed)
- Link B-D
- Q-S (listed)
- Link C-D
- R-S (listed)
- Links preserved
- 1 + 1 + 1 + 1 = 4 of 4
Under A → P, B → Q, C → R, D → S: A-B becomes P-Q; A-C becomes P-R; B-D becomes Q-S; C-D becomes R-S. Score 1 for each listed link: 1 + 1 + 1 + 1 = 4 of 4. Every link arrives at a listed link, so this pairing preserves every link.
Use the idea
When two diagrams claim to show the same connections, write the pairing first, then check each link in a small table instead of trusting how the shapes look.
Where the conclusion applies
Simple undirected networks: no repeated links and no link from a vertex to itself. Matching counts of vertices and links never checks a particular pairing. If line length or direction carried meaning, links alone would not be enough.
Check your understanding: Under A → P, B → S, C → R, D → Q, what does A-B become, and is it listed?
Chapter 4 source: section "Match the links, not the pose". Demonstration C04-D01.
Demonstration 2 of 4
One failed link is a counterexample
If every count agrees, can a pairing still fail?
The highlighted link on the left is carried to the right. If its image is not one of the diamond's links it is drawn dashed: that single link is the counterexample. Exchanging the destinations of C and D repairs the pairing.
\[A-C\]
\[P-S\]
A-C is a link in the first drawing. Under C → S, its endpoints become P and S, written P-S. The diamond's listed links are P-Q, P-R, Q-S and R-S.
Predict first. Under the swapped pairing, follow A-B instead of A-C. Does that link survive, and does its survival rescue the pairing?
Choose an example
Constructed example: the chapter's worked example with C → S and D → R, and its repair.
Calculated values
- Pairing
- A → P, B → Q, C → S, D → R
- Link followed
- A-C becomes P-S
- In the target list?
- no, a counterexample to "every link is preserved"
- Vertices and links
- 4 and 4 in both drawings
- All links preserved
- 2 of 4
Under the swapped pairing (C → S, D → R), A → P and C → S, so A-C becomes P-S. The counts still agree: 4 - 4 = 0 vertices apart and 4 - 4 = 0 links apart. One failed link is enough to refute "every link is preserved" for this pairing. It does not prove the networks differ.
Use the idea
Keep the claim narrow. A failed pairing says this assignment does not work; it does not say the two diagrams have different structures.
Where the conclusion applies
The networks from the opening stay unchanged. Ruling out every possible pairing would take many checks; one failed link rules out only the pairing being tested.
Check your understanding: Under the swapped pairing, what does B-D become, and is it listed?
Chapter 4 source: section "Worked example: the counts agree but this mapping fails". Demonstration C04-D02.
Demonstration 3 of 4
What a reflection keeps and what it moves
When a row of cells is reflected left to right, which features survive?
The top row is the original and the bottom row is its reflection. The same two questions are asked of both rows: how many black cells, and what color is leftmost.
\[\text{black} \mid \text{white} \mid \text{white}\]
Each row has three cells, read left to right relative to the page. A left-right reflection reverses the order. An invariant is a feature that stays the same while something else changes.
Predict first. Choose white-black-white. After the reflection, is the leftmost color still the same? Is the number of black cells?
Choose an example
Constructed example: the chapter's black-white-white row from Figure 4.2, plus three other rows.
Calculated values
- Before
- black-white-white
- After reflection
- white-white-black
- Black cells
- 1 before, 1 after (preserved)
- Leftmost color
- black before, white after (changed)
Reflecting left to right reverses the row: black-white-white becomes white-white-black. Black cells: 1 before and 1 after, and 1 - 1 = 0, so the count is invariant. Leftmost color: black before, white after, so it is changed.
Use the idea
Before saying two pictures are the same, name the change and the feature you are checking, then check that feature before and after.
Where the conclusion applies
Left means left on the page. A reflection never changes a cell's fill or the number of cells, so the black count is always invariant here; the leftmost color can go either way.
Check your understanding: A row black-black-white is reflected left to right. What is the new row, and what is its black count?
Chapter 4 source: section "What survives a reflection". Demonstration C04-D03.
Demonstration 4 of 4
An invariant is not yet a symmetry
The black count survives the reflection. Does that make the reflection a symmetry of the row?
The same reflection is judged by three standards. The frame and the count survive for any row. The full arrangement survives only when every position matches its mirror position, which happens for white-black-white.
\[\text{white} \mid \text{white} \mid \text{black}\]
A symmetry is a transformation that leaves a stated object unchanged in a stated way. The standard says what counts as unchanged: the bare frame, the black count, or every filled position.
Predict first. Keep black-white-white and switch the standard from the black count to the full filled arrangement. Does the answer change?
Choose an example
Constructed example: the chapter's black-white-white and white-black-white rows under three standards.
Calculated values
- Row
- black-white-white becomes white-white-black
- Standard for unchanged
- the full filled arrangement
- Check
- positions that match: 0 + 1 + 0 = 1 of 3
- Unchanged under this standard?
- no
Standard: the full filled arrangement. Check: positions that match: 0 + 1 + 0 = 1 of 3. Not every position matches, so the reflection is not a symmetry of this filled row, even though the black count is invariant.
Use the idea
When someone says two things are the same, ask for three items: the object, the allowed change, and the standard for unchanged.
Where the conclusion applies
A three-cell row and a left-right reflection only. Leaving out the fills is allowed only when the question is about the frame; once fills matter, they must be checked.
Check your understanding: Does the row red-blue-red have left-right reflection symmetry with its fills included?
Chapter 4 source: section "Invariant is not the same as symmetry". Demonstration C04-D04.