Zod is a scientist and inventor, carrying out research and experiments; everyone agrees he is very clever. Zod has two daughters, Molly and Polly, but he dismisses them, insisting that only a son could inherit his genius. His wife Dot gives birth to twins, Joanne and Joseph. Zod takes Joseph and introduces him to his work. In their house are two chambers which the girls are forbidden to enter. They are full of tools and technical equipment; Zod tells the girls they will burst into flames if they go near them. The girls are taken to the kitchen or taught to sew by their mother. Molly and Polly ask their father for a job, but he rejects them, so they leave home. Meanwhile, as the twins grow up, they are indistinguishable but for their clothes. Unknown to their parents they secretly switch places, so that both can learn about everything.

The twins want to be scientists. Zod invites them to his factory. They ask how and why he is making certain things; Zod replies because he can. Joanne hears a cry and finds a guinea pig in a cage; she sets it free. Zod gets angry, and sends Joanne to Professor Yaga, saying that she can only return when her name appears in a scientific journal.

She sets off on her journey. First Joanne goes to college, but she is not made welcome. Pictures of great male scientists line the corridors and the lectures are patronising. However, she completes her studies and continues her journey.

On the way she meets her sister Molly, now a maintenance engineer at the hospital. Molly takes Joanne to the edge of the technological jungle. Initially it is straightforward but as the technology becomes more advanced the going gets harder. She meets an old woman, the Mother of Invention, who shows her the right path to Professor Yaga's laboratory. He is welcoming and says its a pity more women don't apply. The other scientists are not as helpful, one says that she will leave to have children, another asks her to make tea and comments on how nice looking she is.

One day, Joanne makes a discovery, but when it is published her name does not appear. She takes the matter up with Professor Yaga, who replies that there are no great women scientists. Joanne begins to argue, listing examples, but Yaga challenges her to transform a room full of straw into a complexed polymer by the following morning.

During the night, seven women scientists appear to Joanne. One is her sister Polly. They have been rendered invisible by having their work ignored by their male colleagues. Joanne suggests they work together, and they assign themselves different roles. They assemble a machine that transforms the straw into cloth and then into white coats, which Joanne gives to the women scientists, that restores their visibility. Yaga is caught trying to steal their formula, but Joanne explains that he needn't steal it, just as long as they are given credit. He refuses and falls into the machine which transforms him into a beast. From then on the women run the department as a team. Back in their home, Zod and Joseph read about Joanne's research. Her mother, Dot, has begun an Open University course in astrophysics, and when Joanne returns home Zod is cooking.