The evolutionary mechanisms underlying the maintenance of invariant traits are poorly understood, partly because the lack of variance makes these mechanisms difficult to study. Although the number of cotyledons that plant species produce is highly canalized, populations of plants frequently contain individuals with abnormal cotyledon numbers. In a garden study with 1857 wild radish plants from 75 paternal half-sibling families, 89 (almost 5%) had cotyledon numbers less or greater than two. We found evidence for direct selection on cotyledon number, but no evidence for additive genetic variation for cotyledon number. In spite of the very large sample size, our power to detect variation and selection was hampered by the small number of individuals (10) producing more than two cotyledons. Thus, our results provide support for both a lack of genetic variation and selection as reasons for the current lack of variation in wild radish cotyledon number.