Spires are spiral by design, and words that contain this geodesic are rare. You can breathe easy - respiration says so.
"Geodesic" might seem like an ambiguous term here, but bear with the definition as on dictionary.com:
"Noun. A curve that locally minimizes the distance between two points on any mathematically defined space, such as a curved manifold. Equivalently, it is a path of minimal curvature. In noncurved three-dimensional space, the geodesic is a straight line."
Spires (architecturally) are typically considered to comprise straight lines from a collection of points (at the base) to a single point (at the apex) - a spiral is comprised of curves denoting the shortest possible path from one point to the next traversing the circumference at any given point in the geometry of the spire. A differential addressed by WikiDiff thus:
"... the difference between spiral and spire is that spiral is (geometry) a curve that is the locus of a point that rotates about a fixed point while continuously increasing its distance from that point while spire is (geometry) the part of a spiral generated in one revolution of the straight line about the pole.
As nouns the difference between spiral and spire is that spiral is (geometry) a curve that is the locus of a point that rotates about a fixed point while continuously increasing its distance from that point while spire is or spire can be one of the sinuous foldings of a serpent or other reptile; a coil.
As verbs the difference between spiral and spire is that spiral is to move along the path of a spiral or helix while spire is of a seed, plant etc: to sprout, to send forth the early shoots of growth; to germinate or spire can be (obsolete) to breathe.
As a adjective spiral is helical, like a spiral."
So the two definitions seem related but confusing, although it could easily be said that "spiral" is the collective noun for "spire".
The seedling uncoils from a spiral, developing in a Fibonacci sequence of cellular repetitions to produce its adult form. Notwithstanding that all this is metaphorically describing something we already understand. Like the helix of that DNA strand we've never seen other than in CGI...
At the top of any apex is a singular point at which you can say, "there it ends". There may well be a new beginning after the point of completion, of course, but that phase of life is over, as it were; the spire of your aspiration has reached the singularity where the laws of physics break down.
Inspiration being the fuel that gets us there, it stands to reason (if you credit language with any synchronicity in its creation, that is) that "Insp" is 'within' and "Asp" is 'without', or 'lies back of' as Florence Scovel Shinn put it. Look her up via the link - she was way ahead of her time.
Let's for a moment let nothing be coincidence. That DNA spiral repeated throughout your physical form, it lets us know that the foundations of nature - including our own composition - rely on spiral networks to exist. In science you're likely to find a similar word referencing all kinds of variants - the word is "Chiral". scroll through the list here in Wikipedia and the correlations become very clear. Better still, read the brilliant article by Philip Ball, linked to the image of bees above.
Exploring the foundations of Quantum Mechanics, you'll find it opens doors to new appreciations and ways of thinking that prove extremely useful over time. Arguments over choice v. destiny will rage on no doubt, but at the end of the day it's your life, my life, all lives with those chiralities to contend with. We can always start again, any time we like.