How could the shape of extra dimensions affect the masses and charges of particles?

In string theory, the fundamental building blocks of the universe are tiny, one-dimensional strings. The mass and charge of a particle are determined by the specific vibrational pattern of the string, much like how different vibrations of a violin string produce different musical notes.1

Because strings are infinitesimal, they vibrate within the extra spatial dimensions required by the theory. These dimensions are postulated to be "compacted" or "curled up" at scales too small to be detected, similar to how the circular cross-section of a garden hose might appear as a one-dimensional line from a distance.12

The exact geometric shape and size of these extra dimensions act as a constraint on the strings' movement. Just as the physical structure of a French horn influences the vibrations of the air passing through it, the geometry of the extra dimensions dictates how the strings vibrate. Consequently, because these vibrations define a particle's properties, the specific geometric form of the extra dimensions directly determines the masses and charges of the particles observed in our four-dimensional space-time.1