Has anyone claimed that beauty is order, or anything close to that?
Seems like quite a classical idea, but I don't know of any philosophers who say this.
Aristotle, in The Poetics, comes to mind.
Again, a beautiful object, whether it be a living organism or any whole composed of parts, must not only have an orderly arrangement of parts [beginning middle and end?], but must also be of a certain magnitude; for beauty depends on magnitude and order. Hence a very small animal organism cannot be beautiful; for the view of it is confused, the object being seen in an almost imperceptible moment of time.
There are many classical conceptions like that.
In a way also Hume, but beauty as an order of parts that gives pleasure/satisfaction.
...we shall find that all of them resolve into this, that beauty is such an order and construction of parts, as either by the primary constitution of our nature, by custom, or by caprice, is fitted to give a pleasure and satisfaction to the soul. This is the distinguishing character of beauty, and forms all the difference betwixt it and deformity, whose natural tendency is to produce uneasiness.
A Treatise of Human Nature - Hume