THEANO (born c. 546 B.C.), the wife of the Greek mathematician and philosopher Pythagoras, ran the Pythagorean school in southern Italy in the late sixth century B.C. following her husband’s death.She is credited with having written treatises on mathematics, physics, medicine, and child psychology.It is said that she formulated the theory of the “Golden Section” as well as the theory of the “Harmony of the Spheres”.Theano’s husband, Pythagoras (c. 582-500 B.C.), was inspired one of the most influential sects in the ancient world.Best known for devising the Pythagorean Theorem—which states that the sum of the squares of the sides of a right triangle is equal to the square of the hypotenuse—Pythagoras was considered the greatest scientist of antiquity by classical Greek scholars and is considered to have been the first mathematician.However, given that Pythagoras lived seven generations before Plato, most of the information about him comes from fairly late sources—a few as late as the third century A.D. Another problem is that some of these sources are of doubtfulreliability.

She is said to have published passages from Pythagoras on the “number theory” of the “Golden Section” as well as the theory of the “Harmony of the Spheres”.
Theano is mentioned by Athenaios, the Souda dictionary, Diogenes Laertius and Iamblichus.
The common opinion today is that we can attribute the “Pythagorean Apothegmata”, “Advice to Women”, “On Piety”, “Philosophical Commentaries”, “Epistles”, “Life of Pythagoras” to Theano. Her greatest contribution, however, is said to have been the publicizing of the principle of the Golden Ratio used in architecture by the ancient Greeks as well as the ancient Egyptians. The formulation of the theory of the Golden Section (of the number φ) is similar to the formulation of the theory of the constant π created by Pythagoras.

References to Pythagoras’s ideas can be found in earlier writings, including those of Empedocles, Heraclitus, Herodotus, Plato, and Aristotle.https://biography.yourdictionary.com/theano 

In her treatise On the Universe she formulated and published the Pythagorean theory that the universe consists of 10 concentric spheres moving around a central fire: the Sun, the Moon, Saturn, Jupiter, Mars, Venus, Mercury, Earth and the anti-Earth. There are also fixed stars. The distances between the spheres and the central fire have the same proportions as the intervals of the musical scale. Theano and her daughters were also considered important physicians.
Diogenes Laertius mentions that Theano once advised women when they meet their husbands to take off their morality along with their clothes. This was a reference to Herodotus’ dictum that “a woman sheds her morals when she sheds her clothes”, which was made very popular in the early Christian years and the Middle Ages by religious thinkers. But Plutarch who was a moralist had already expressed his objection to this saying saying “she who is moral is clothed in her morality even when her clothes are thrown aside”. In addition, when Theano was asked which clothes she was referring to she replied “those that make you a woman”. The concept of the morality of both women and men was vital to Pythagoreanism and naturally influenced the thinking of Theanos. It is said that she once showed her arm as she tried to arrange her clothes and someone said “what a beautiful arm!” Then she replied “yes, but it is not a public good”. This incident shows on the one hand that a woman’s bare arm could be considered erotically provocative and on the other hand that the Pythagoreans placed strict limits on sexuality and provocation.

The school of Pythagoras existed for at least 200 years after his death. For centuries historical researchers have underestimated the role of women in relation to the ancient sciences. Usually, they were seen through the prism of “being in the shadow of their husbands”. But the history of the Pythagorean school showed that women were not only great scientists but also inspired many men in the field of science.
Theano was one such woman and died around the 5th century. e.g. Also, it is probable that her works and research were known (in the sense that they were studied) by other women who followed the path of science during ancient times. Although scholars have traditionally overlooked and underestimated the contribution of women to history and science, Theanou’s work was known to her contemporaries and was a source of inspiration for many men and women. Among those who were influenced and inspired by Theano is believed to be Hypatia the Alexandrian mathematician and philosopher, who unfortunately met a horrible end.

what wikipedia mentions about Theano, regarding the theories attributed to her, are not based on ancient sources, but only on American fiction books. see what wikipedia mentions about Theano, regarding the theories attributed to her, are not based on ancient sources, but on American fiction books only. see Plant, Ian Michael (2004). Women Writers of Ancient Greece and Rome: An Anthology. University of Oklahoma Press. P. 68.,69, ISBN 978-0-8061-3621-9. and Mary Ellen Waithe, A History of Women Philosophers. Volume 1, 600 BC-500 AD

Excerpt from the book: “Pythagoras… Secret Teaching & Modern Science”, chapter, Pythagorean Women Philosopher