Thomasina speaks the very first line of the play, a line that tells us from the start that Arcadia may be set in Regency England, but it's no Jane Austen novel:
THOMASINA: Septimus, what is carnal embrace? (1.1)
First, while this question begins the play on a humorous and slightly shocking note, it also neatly demonstrates that Thomasina is embarked on the transition from childhood to adulthood, in the limbo of adolescence. She's obviously picked up the term "carnal embrace," which shows some exposure to sexual knowledge, from somewhere, but she doesn't know enough to know what it means, so she has to ask. She can ask this question only because she's on this border: if she knew more, she wouldn't need to ask the question, and if she knew less, she wouldn't know there was a question to ask.
And sex isn't the only thing Thomasina has questions about. There's also Newtonian physics:
THOMASINA: When you stir your rice pudding, Septimus, the spoonful of jam spreads itself round making red trails like the picture of a meteor in my astronomical atlas. But if you stir backward, the jam will not come together again. Indeed, the pudding does not notice and continues to turn pink just as before. Do you think this is odd? (1.1)
This passage gives us an idea of how Thomasina thinks: she puts abstract ideas in the context of familiar objects (the laws of physics as rice pudding) and she's able to make connections between the very small and the very large (pudding and meteors). In the scientific context as well as the sexual one, her almost-but-not-quite knowledge means that she asks questions neither a child nor an adult would think to ask.
The play uses Thomasina's questions about sex and physics to portray her in that stage between ignorance and knowledge, innocence and experience. How does she go about gaining knowledge? How does her in-the-middle-ness – her remaining childhood imagination and innocence – help her to see more than the adults around her?
Geometry Stinks, But Not For The Reasons You Think It Does
The more knowledge Thomasina gains, the more she hates its limitations. Geometry is a particular target of her wrath:
THOMASINA: Each week I plot your equations dot for dot, xs against ys in all manner of algebraical relation, and every week they draw themselves as commonplace geometry, as if the world of forms were nothing but arcs and angles. God's truth, Septimus, if there is an equation for a curve like a bell, there must be an equation for one like a bluebell, and if a bluebell, why not a rose? (1.3)
While Septimus is content to have two worlds – geometry and nature – divided up like girls and boys at a middle school dance, Thomasina wants the two mixed. What drives Thomasina's assertion that an equation "must" exist? Is she pulling a Bernard and saying that she just knows, or is she using the powers of reason? And does that borderline position discussed above affect how she's thinking about these questions? How?
Waltzing: It's All Innocent, Right?
While the sixteen-year-old Thomasina who appears later in the play is still interested in "the action of bodies in heat" (2.7; we'll leave it as an exercise for the reader whether that's a question of sex or science), it's waltzing that's first and foremost in her mind. Her concern with learning the latest dance craze develops her character's borderline status between innocence and experience in a new direction.
When Thomasina comes downstairs late at night to claim her promised waltz lesson from Septimus, she's surprised that her tutor doesn't immediately understand that her kiss means "teach me to waltz now" – as opposed to "I want to have sex with you." She tells him, "Do not act the innocent! Tomorrow I will be seventeen!" (2.7). Her words are unintentionally funny: she's thinking about dancing in the vertical, while his thoughts are more of the horizontal variety. Yet he's the one being accused of "acting the innocent." It's a funny turn of phrase – but it also suggests that innocence is more complicated than it seems.
Things only get more complicated when there's more kissing, instigated by Septimus this time, and Thomasina invites Septimus up to her room. This all happens so quickly, and so close to the end of the play, that it's uncertain what Thomasina's motives are. Has she been harboring a secret crush on Septimus? Is she hoping he'll be an adequate stand-in for her Byron fantasy? Thomasina's abrupt death, and the end of the play, mean that we'll never know – and that, as a character, Thomasina is forever poised on that boundary between innocence and experience.