Gregor Mendel was an Austrian monk in the 19th century who uncovered many of the basic principles of inheritance and genetics by his experiments with pea plants. By observing several generations of the plant, he could determine which traits were dominant and which were recessive, as well as identify several patterns of inheritance. He mated different varieties of plants, using a brush to transfer pollen. Mating different varieties, or crossing different genotypes, of a plant is called hybridization.

Mendel started his experiments with purebred plants, that is, identical plants that when mated had offspring with traits identical to their parents (such as having all purple flowers). Purebreds are homozygous for the trait that they exhibit, such as flower color. They may be homozygous dominant (PP=purple flowers) or homozygous recessive (pp=white flowers). When Mendel crossed purebred purple plants with purebred white plants, all plants in the first generation of offspring were purple. The punnett square for that cross is as follows, using the letter p to represent alleles for flower color:

Table 1. Punnett square for monohybrid cross of two purebred plants.

When breeding experiments only follow one trait, such as flower color. They are called monohybrid crosses. As you can see from the punnett square above, all offspring produced were heterozygous for flower color. Because they had one dominant allele (P) in their genotype, they all exhibited the dominant purple phenotype. However, when Mendel crossed the heterozygous offspring of two different purebreds, he observed a different pattern of inheritance in this second generation of offspring.

Table 2. Punnett square for monohybrid cross of two heterozygous parents.

The second generation of offspring had the following genotypes: PP, Pp, Pp and pp. Instead of having all purple flowers, some offspring had white flowers. Those with a dominant allele (P) had purple flowers, while those with two recessive alleles (pp) had white flowers.

A ratio is a mathematical expression to describe the proportion of different items in a set. If there are 3 boys and 2 girls in a room, the ratio of boys to girls in the room is 3:2. Similarly, the ratio of purple to white flowers, or phenotypes, in the offspring of the heterozygous cross is 3:1. The ratio of genotypes in the offspring is slightly different. Since there were 3 different genotypes in the offspring (1 PP, 2 Pp and 1 pp), the ratio of genotypes is 1:2:1.

Mendel's breeding experiments became more complicated when he performed crosses to observe more than one trait at once, such as pea color and texture, called dihybrid crosses. Punnett squares for dihybrid crosses are larger. The following will illustrate an example of the punnett square resulting from a dihybrid cross of pea plants, with both parents being heterozygous for pea color (Yy) and pea texture (Rr).

Dominant pea color is yellow (Y), while recessive color is green (y). Dominant pea texture is round (R), while recessive is wrinkled (r). The genotype for both parent pea plants is YyRr. The first step to make this punnett square is to determine the combinations of alleles that each parent can give separately. To do this it's necessary to use a technique from algebra called F.O.I.L. (first-outer-inner-last) on the genotype YyRr.

The genotype for trait one (pea color) is Yy, while the genotype for trait two (pea texture) is Rr. First, multiply the first letter of each trait's genotype (Y*R), then the two outer letters (Y*r), then the two inner letters (y*R) and finally the last letters (y*r). These are the four possible combinations of alleles that each parent plant can donate to offspring. To determine the offspring, you cross multiply all four parent alleles. As you can see from the punnett square below, a dihybrid cross results in many more possibilities for the genotype and phenotype of the offspring.

Table 3. Punnett square for dihybrid cross of two parents heterozygous for both traits.