I have seen the angle referred to as being dimensionless, but I am seeing it as phase of cycle. For example, if an angular width is given, it is the phase distance from one side to the other.

You're touching on a real and often glossed-over subtlety in how angles are treated in physics and mathematics.

The standard claim: angles are dimensionless

In the SI system, the radian is defined as a derived unit with dimension 1 (i.e., dimensionless). The reasoning is:

θ = \frac{s}{r}

where $s$ is arc length and $r$ is radius. Both have dimension of length, so their ratio is a pure number. Thus "angle" carries no independent physical dimension — it's length/length.

This is why you'll see angular quantities plugged into exponentials, sines, etc., and why $ℏ$ has units of action (J·s) rather than action per radian.

Your view: angle as phase of a cycle

What you're describing is arguably the more operationally meaningful picture, and it's very common in physics:

A full rotation = one full cycle = $2 π$ .
An angular width $Δ θ$ is literally how far apart two things are in the cycle.
In harmonic motion, $ω t + ϕ$ is explicitly called the phase, and it's measured in radians.
In interference/diffraction, phase differences $δ = \frac{2 π}{λ} d \sin θ$ are angular in nature.

So when you say "angular width is the phase distance from one side to the other," you're identifying angle with position within a periodic cycle — which is exactly what phase is.

Why both views coexist

The tension resolves like this:

Dimensional analysis view: Angle is a ratio of lengths, hence dimensionless. This is convenient for unit consistency (e.g., $\sin θ$ requires a pure number).
Phase view: Angle is a coordinate on a circle (or on the universal cover, the real line mod $2 π$ ). It parametrizes a cycle. This is what you actually measure and what appears in $e^{i θ}$ .

These aren't contradictory — they're two descriptions of the same mathematical object. The radian is the natural "unit" that makes the identification $θ \leftrightarrow$ phase explicit, because:

e^{i θ} = \cos θ + i \sin θ

and the periodicity $e^{i (θ + 2 π)} = e^{i θ}$ is exactly the statement that $θ$ lives on a cycle.

A useful way to hold it

Dimensionless = it's a pure number (no meters, no seconds).
Phase = what that pure number means: a location in a cycle.

The dimensionless label tells you it won't mess up your units. The phase interpretation tells you why it appears in oscillatory, rotational, and wave phenomena the way it does.

So your intuition isn't wrong — it's actually the more physically transparent reading. The "dimensionless" statement is a statement about units, not about meaning. Angle-as-phase is the meaning.

User:

I would like to abstract the concept of phase to its bare root, then apply it to each aspect of a temporal object as an imaginary axis: A drinking glass, a glass for drinking, for example serves only as an actual glass for drinking when various of its aspects are aligned. It must be in a clean phase, it must be in a handy phase, and so on.

DeepSeek:

Phase as bare root: alignment of aspects

Stripping phase to its core

If we remove the circle, the radians, the sine waves, what remains at the root of "phase" is this:

A periodic variable's current position in its cycle.

But even more primitively, phase is a relation between two or more cyclic processes — it answers: where is A relative to B in their respective cycles?

In phase: peaks align with peaks.
Out of phase: peaks align with troughs.
Phase offset: a fixed displacement between them.

So the bare root of phase is not "angle" and not "cycle" alone, but:

A condition of mutual alignment among variables that each move through their own cycles.

Applying this to a temporal object

A drinking glass, as you say, is not simply a glass. It is a bundle of aspects, each with its own cyclic or quasi-cyclic behavior:

Aspect	Its cycle
Cleanliness	dirty → washing → clean → use → dirty
Location	cupboard → hand → table → hand → cupboard
Availability	reserved → free → in use → free
Temperature	ambient → cooled → warming → ambient
Structural integrity	intact → stressed → intact or broken
Social role	stored → offered → drunk from → cleared

Each aspect is a phase variable with its own period, its own waveform, its own current position.

The imaginary axis move

Here is where your framing becomes powerful. For each aspect, define an imaginary axis — not imaginary in the sense of "unreal," but in the sense of orthogonal to the actual:

The actual glass is what is present right now.
Each imaginary axis is a dimension along which that actual glass must be phase-aligned with the others for it to function as "a glass for drinking."

So the glass-as-drinking-vessel is not a substance. It is an intersection condition:

Drinkable glass = ⋂_{i} {{aspect}_{i} is in the required phase}

If cleanliness is in the "clean" phase, location in the "hand" phase, integrity in the "intact" phase, temperature in the "acceptable" phase — then the glass is a drinking glass. If any one drifts out of phase, the object fails its role.

Why "imaginary axis" is the right metaphor

In complex analysis, a real quantity becomes richer when paired with an imaginary component: $z = x + i y$ . The imaginary part doesn't negate the real; it adds a dimension of variation that the real alone cannot express.

Likewise:

The real axis of the glass is its bare physical presence — matter occupying space.
The imaginary axes are its aspects — each a phase dimension along which the object must be aligned.

An object becomes functional when its imaginary components are synchronized. It becomes dysfunctional when they decohere.

Phase alignment as the condition of actuality

This gives a clean ontology:

A thing is a bundle of aspects.
Each aspect is a cyclic variable with a phase.
Function is phase alignment across aspects.
Failure is phase misalignment.
The "actual" object — the drinking glass as such — is not a static substance but a standing phase relation.

A glass in the cupboard is still a glass materially, but its drinking-glass phase is not active. Its location aspect is out of phase with its use aspect. The object exists, but its role does not.

Generalizing

This applies to anything temporal:

A conversation is actual when turn-taking, attention, topic, and mood are phase-aligned.
A tool is usable when location, readiness, and task are aligned.
A person is "present" when attention, intention, and situation are aligned.
A signal carries information when its phase relationships are preserved.

In every case, the entity is not a thing but a phase-locked bundle of aspects. The "imaginary axes" are the dimensions along which locking must occur.

The bare-root formula

Phase = the alignment condition among cyclic aspects that allows a bundle to function as a unity.

Or more tersely:

Phase is the grammar of co-occurrence.

Your drinking glass is a drinking glass only when its aspects are in phase. Otherwise it is just glass — matter waiting for alignment.

The Root of Phase