What were Mendel's findings regarding patterns of inheritance?

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Gregor Mendel was an Austrian monk who experimented with pea plants to learn more about patterns of inheritance. Today, many regard him as the father of the study of genetics, and we call the patterns of inheritance he described as Mendelian inheritance or Mendelian genetics. 

Mendel's pea experiments were based on his selective pollination of several plants. He observed the frequency of different phenotypes like purple or white blossoms, yellow or green seeds, and wrinkled or round seeds. Mendel discovered that by breeding certain plants together, the offspring might be more likely to resemble just one of the parent plants. He also discovered that there were no intermediate forms for the phenotypes — a yellow pea plant and a green pea plant do not produce a yellow-green offspring plant, for example.

Mendel developed two laws or principles of inheritance which informed later studies of genetics. The first law is the law of segregation, which states that organisms have two alleles for a particular trait. During reproduction, these two alleles separate from one another, and just one is passed on to the offspring. The second law is the law of independent assortment, which states that alleles are inherited independently of one another. For example, a parent plant which has purple flowers and yellow peas does not necessarily pass on both of these traits to the offspring. In humans, we can think of this in terms of the relationship between hair and eye color — just because someone inherits brown hair from their mother, it does not necessarily mean they will also inherit her blue eyes.

Mendel also used Punnet squares to predict or describe the potential genotypes of a population, the corresponding phenotype, and the frequencies of the related traits. This web page does a great job of giving a more in-depth description of how to create and interpret Punnet squares. Mendel used Punnet squares and selective breeding of his pea plants to determine that there are both dominant and recessive traits at work.