How does the garden-hose analogy explain hidden dimensions in string theory?

The garden-hose analogy, rooted in the work of Theodor Kaluza and Oskar Klein, illustrates how string theory accounts for extra spatial dimensions that are not immediately apparent to human observation.1

The analogy compares dimensions to the physical structure of a long garden hose:
* Large dimensions: From a distance, a long garden hose appears to be a one-dimensional line. This represents the three dimensions of common experience, which are large and manifest.1
* Hidden dimensions: Upon closer inspection, the hose has a circular cross-section, providing a second, smaller dimension. If this circular girth were shrunk to an infinitesimal size, it would become invisible to the naked eye, leading a casual observer to perceive only the one-dimensional length.1

Similarly, string theory posits that while three spatial dimensions are large, the other six are "crumpled" or rolled up so tightly—at approximately ${10}^{-33}$ cm—that they evade detection. These hidden dimensions are critical because strings vibrate within them; their specific geometric form influences how strings vibrate, which in turn determines the fundamental properties of particles, such as mass and charge.1