# Geometry applied to breeding common beans (Phaseolus vulgaris)

### Abstract

The primary components of common bean (*Phaseolus vulgaris*) grain yield (W) are the number of pods (X), the number of grains per pod (Y), and the weight of the grains (Z). In 1964, Grafius suggested using geometry in plant breeding; W corresponds to the volume of a parallelepiped with three axes, X, Y, and Z. Because the cube is the largest parallelepiped by volume, maximum yield is obtained when the relative contributions of X, Y, and Z are the same. We evaluated individual plants of a ‘Talismã’ x ‘L.59583’ cross in two sowing periods. The sum of squares of deviations from the ideal plant (GI), i.e., the plant in which the X, Y, and Z contributions were the same, was estimated. Mean and variance genetic components, and genetic and phenotypic correlations between the characteristics were also estimated. Good concordance was observed in the magnitude and direction of the genetic and phenotypic correlation estimates of the paired characteristics. However, a low GI heritability (*h** ^{2}_{r} *= 6.7%) indicated that success due to selection should be small. Ninety-four progenies of ‘Pérola’ x ‘ESAL 686’ crosses were also evaluated, where X, Y, Z, and W were obtained and GI was estimated. The

*h*

*estimate was higher, but still low (*

^{2}*h*

*= 39.0%). Therefore, the selection of individuals to obtain plants in which the X, Y, and Z products tend to the cube is unfeasible, because the sums of X, Y, and Z vary between individuals. In addition, the GI*

^{2}*h*

*2*value was low.

The primary components of common bean (*Phaseolus vulgaris*) grain yield (W) are the number of pods (X), the number of grains per pod (Y), and the weight of the grains (Z). In 1964, Grafius suggested using geometry in plant breeding; W corresponds to the volume of a parallelepiped with three axes, X, Y, and Z. Because the cube is the largest parallelepiped by volume, maximum yield is obtained when the relative contributions of X, Y, and Z are the same. We evaluated individual plants of a ‘Talismã’ x ‘L.59583’ cross in two sowing periods. The sum of squares of deviations from the ideal plant (GI), i.e., the plant in which the X, Y, and Z contributions were the same, was estimated. Mean and variance genetic components, and genetic and phenotypic correlations between the characteristics were also estimated. Good concordance was observed in the magnitude and direction of the genetic and phenotypic correlation estimates of the paired characteristics. However, a low GI heritability (*h** ^{2}_{r} *= 6.7%) indicated that success due to selection should be small. Ninety-four progenies of ‘Pérola’ x ‘ESAL 686’ crosses were also evaluated, where X, Y, Z, and W were obtained and GI was estimated. The

*h*

*estimate was higher, but still low (*

^{2}*h*

*= 39.0%). Therefore, the selection of individuals to obtain plants in which the X, Y, and Z products tend to the cube is unfeasible, because the sums of X, Y, and Z vary between individuals. In addition, the GI*

^{2}*h*

*2*value was low.