We perceive our surrounding as a 3 dimensional world. Geometry is part of math which are a collection of abstract concepts that arose in our mind. Most of the things around us can be described in terms of geometry. Why there is geometry in nature ? Or what gives the universe a geometry?
Some philosophers may tell you that there isn't geometry in nature, and that your mind imposes a geometry, allowing you to see beyond fragments of sense data to a gestalt.
Now, although we may have the impression that our perception of space depends upon only the functioning of our minds and organs of perception, and that space itself simply exists and requires no process to maintain, some theological views assert otherwise. For example, it has been said that it requires the active will of God to maintain the universe in existence.
Now, although seeing yourself in a mirror, and reading about the physiology of the eye may persuade you that we understand the true nature of space, is there any reason to believe that you wouldn't perceive such things if your mind were interacting with a simulation of physical reality? Alternatively, what if there is no underlying physical universe as a basis for a simulated reality, and it is the active creation of your perceptions that maintains the appearance of your interaction with a physical universe? In that case, a miracle wouldn't require God's intervention in a self-regulating physical universe. God could provide you with a gift-wrapped present, and God wouldn't have to decide what is inside until somebody opens the wrapping.
Problems in architecture, surveying land, map-making, and navigation helped to motivate the development of geometry. We have no evidence that such a simple thing as Pasch's postulate was discovered until after a series of civilizations had come into existence and then faded away. Although any given view regarding the ultimate relationship between human beings and geometry may depend upon metaphysical presuppositions, it seems to be beyond dispute that our understanding of geometry has been a byproduct of the formation and development of civilizations -- and competition between rival civilizations -- over a vast period of time.
I'm assuming that you presume that - since it is an abstract and human-made system - it is counterintuitive to find (traces of) geometry in nature.
But geometry (as well as mathematics and physics) has its fundament in nature (or reality). What you regard as geometry in nature is a natural occurrence interpreted as belonging to an abstract order.
It's in our human nature to label and categorize everything, but it comes with that hazard of conflating cause and effect.
To illustrate, compare your deduction to measuring the speed of objects falling down: how come that speed corresponds to g (9.80665 m/s2)?
Every formal system of geometry is in 1:1 correspondence with a formal system of algebra. Euclidean geometry in 3D is tied to the (ordinary) algebra of quantification that we use to make sense of the world we inhabit. Riemannian geometry and the mathematics which underlie it describes noneuclidean geometry, one expression of which is connected to general relativity- which describes how euclidean geometry is altered in the presence of concentrations of matter or energy, thereby giving rise to gravity.
In this sense, the fundamental postulates of general relativity furnish a particular geometric structure to the universe.
Curvature. Look at the definition of any object and we are left with curvature. Take a glass for example. Does the glass exist becomes of the particles? Or does it exist because it has shape?
Assuming particles we are still left with the particle being 99.99...percent formless (or empty). What we understand of all phenomenon are strictly forms.
This of course goes back to Plato's theories of forms, but it reflects the medieval/socratic/presocratic view of the "Circle" (or sphere by default) being divine.
Take any object, pick a spot and trace all the curvature and eventually you are left with a loop...an approximation of the circle. It is one loop existing through many, which reflects some emanation stances that stem from Platonic thought.
It is pretty much just shapes at the end of the day, with the shapes being curvature and thus curvature being just a boundary or limit that is intrinsically empty upon further speculation.
Even the basic Trillema (Munchauseen/Agrippan) references and/or implies basic spatial terms of the point, line and circle. We reason linearly and circulary through the point of view (empty nature of how we assume reality).