Renaud Di Francesco , the shapes and symmetries we observe in nature are intriguing. I have not found any paper on the extensive study of the geometry of the fruit base specifically. Generally, rose family plants (including strawberry) produce flowers with four or five sepals. I think the pentagonal shape of the whitish tissue at the base of the strawberry fruit indicates the attachment points for the five sepals.
Thanks Madhav, you find regular structures in bees as well, to speak of living species. However crystallography as well gives inspiring geometric structures, not to mention the chemical components with carbon atom, methane, benzene, etc... Yes material geometry is seen in nature, and the human construct from Euclid to Lobatchevsky and Bourbaki, Mandelbrot, etc, relies on material experience, as well as axiomatic/ deductive frameworks.
Fascinating world we live in.
It's safer to eat a strawberry pentagon than to eat a benzenic hexagon though (C6H6)...
The answer for the pentagonal shape comes from the number of petals and the calyx. Most strawberries usually have 5 petals to the flower and 10 sepals to the calyx. These mould the strawberry into the shape you saw. I expect many soft fruits will do the same.
Essentially, it is true that whenever we notice an exceptional beauty and harmony, we will usually reveal the presence of golden ratio, i.e. pentagon in this case...