The Collaborative International Dictionary of English v.0.48:

Platonic \Pla*ton"ic\, Platonical \Pla*ton"ic*al\, a. [L.
   Platonicus, Gr. ?: cf. F. platonique.]
   1. Of or pertaining to Plato, or his philosophy, school, or
      opinions.
      [1913 Webster]

   2. Pure, passionless; nonsexual; philosophical.
      [1913 Webster]

   Platonic bodies, the five regular geometrical solids;
      namely, the tetrahedron, hexahedron or cube, octahedron,
      dodecahedron, and icosahedron.

   Platonic love, a pure, spiritual affection, subsisting
      between persons of opposite sex, unmixed with carnal
      desires, and regarding the mind only and its excellences;
      -- a species of love for which Plato was a warm advocate.


   Platonic year (Astron.), a period of time determined by the
      revolution of the equinoxes, or the space of time in which
      the stars and constellations return to their former places
      in respect to the equinoxes; -- called also great year.
      This revolution, which is caused by the precession of the
      equinoxes, is accomplished in about 26,000 years.
      --Barlow.
      [1913 Webster]

The Collaborative International Dictionary of English v.0.48:

Platonic \Pla*ton"ic\, n.
   A follower of Plato; a Platonist.
   [1913 Webster]

WordNet (r) 3.0 (2006):

Platonic
    adj 1: of or relating to or characteristic of Plato or his
           philosophy; "Platonic dialogues"
    2: free from physical desire; "platonic love"

Moby Thesaurus II by Grady Ward, 1.0:

49 Moby Thesaurus words for "Platonic":
   Augustinian, Averroist, Berkeleian, Cartesian, Hegelian,
   Heideggerian, Heraclitean, Humean, Husserlian, Kantian, Leibnizian,
   Neo-Hegelian, Neo-Pythagorean, Neoplatonic, Neoplatonistic,
   Parmenidean, Peripatetic, Platonistic, Pyrrhonic, Pythagorean,
   Sartrian, Schellingian, Schopenhauerian, Scotist, Socratic,
   Spencerian, Thomist, Thomistic, Viconian, Wittgensteinian,
   abstinent, animist, animistic, celibate, continent, idealist,
   idealistic, immaterialist, intact, maidenly, panpsychist,
   panpsychistic, pre-Socratic, solipsistic, spiritualist,
   spiritualistic, vestal, virgin, virginal



The Devil's Dictionary (1881-1906):

PLATONIC, adj.  Pertaining to the philosophy of Socrates.  Platonic
Love is a fool's name for the affection between a disability and a
frost.