Tensions in Practice: A compact count in tension with retained orientation¶
An empty or single-dot two-by-two board
The board is empty or contains one dot in one of four cells. A count reports only 0 or 1. A cell mask records which cell holds the dot. Turn a top-left dot clockwise by 90 degrees: it moves to top right. The count stays 1, while the mask must turn with the board. Both squares agree whichever path is followed, but their output transformations differ.
Discard irrelevant orientation
Return a count that remains unchanged under board rotation.
Preserve a usable location
Return a mask that rotates with the input.
Why these aims pull against each other
An invariant count cannot later locate the dot. Retaining a mask makes the rotation available to later steps but carries more state and a coordinate convention.
Choose an arrangement to see what changes and what remains difficult.
Follow either route around a square: map then transform the output, or turn the input then map. Quarter-turn means 90 degrees clockwise. The count’s output transformation is no change; the mask’s is a matching turn.
What this choice protects
What it costs
When it fits
Compare the arrangements
Return the invariant count
Encode only whether there is a dot. All four single-dot boards map to the same value 1.
- What it protects
- A fixed one-bit result answers presence without a location convention.
- What it costs
- The result cannot identify which of the four cells holds the dot.
- When it fits
- Fits a task whose downstream question is only whether a dot is present.
Illustration note: Invariance is equivariance with an output action that does nothing. The comparison does not claim the two are unrelated mathematical categories.
Return the turning mask
Store four binary cells in fixed row order. Rotating the input rotates those output cells by the same quarter turn.
- What it protects
- A downstream step can identify the location and transform it consistently.
- What it costs
- This declared format stores four bits rather than one and requires a shared cell orientation.
- When it fits
- Fits a downstream task that must act on the dot’s location after rotation.
Illustration note: The fixed four-bit versus one-bit formats are specified choices, not an optimal compression theorem or a general cost claim for every equivariant map.
What this illustration does—and does not—establish
The source supplies the structural tension. This bounded example makes a particular relation inspectable; the aims, conditions and residual costs are part of the comparison.
- Only empty or single-dot boards and quarter turns are admitted. No continuous-rotation or noisy-image guarantee is implied.
- A mask here is just the board’s four binary cells, not a learned vision model.
- The squares express exact transformation consistency. They do not establish that rotation should be ignored for every recognition task.
Source entries
Equivariance
The canonical tension motivates this comparison. The setting and arrangements are declared editorial illustrations, not observed findings.
Equivariance preserves information that invariance discards, but at a cost
Carrying a symmetry forward equivariantly keeps more information available to downstream stages than collapsing it to an invariant immediately, which is often the right design.
The source operation
The output does not stay fixed (that would be invariance) but changes *in lockstep* with the input, so the transformation can be applied before or after the map with the same result.