Geometric series exists in nature: Evidence from sorted area sequences of floral parts and leaves.

The concept of a geometric series (GS) plays an important role in mathematics. However, it has been neglected in describing biological size series. Herein, we show that a GS describes the nonreproductive (perianth) parts of the flowers of four Magnoliaceae species and two Rosaceae species and the leaves of 60 Alangium chinense and 60 Shibataea chinensis shoots. The sorted areas of floral parts and leaves formed a sequence that was fitted by a GS with the mean of the quotients of two adjacent members in the sequence as the common ratio of a GS. The mean absolute percent error (MAPE) was used to measure the goodness of fit of each GS. Over 99.7% of the MAPE values (371 out of the 372 tested flowers) were less than 10%, and over 97.8% of the MAPE values were less than 5%. Likewise, over 77.5% of the MAPE values (93 out of the 120 tested shoots) were less than 10%, and over 35% of the MAPE values were less than 5%. These analyses provide empirical evidence that the GS exists in nature, and confirm the usefulness of a classical algebraic formula for the study of plant developmental biology.