How many dimensions ?

String theories seem to required many additional dimensions. How can this possibly relate to our 4-d reality.

T. Kaluza and by O. Klein (1920's) introduced the idea that our universe might have more than the three familiar spatial dimensions
        Introduced more than half a century before the advent of string theory .
          The basic premise of such Kaluza-Klein theories is that a dimension can be either large and directly observable or small and essentially invisible.
             An analogy with a garden hose can be helpful


kauza_klien




kaluza_kliens two extended dimensions and two compactified dimensions



For string thoery the familiar 3 -spatial dimensions are revealed when six of the dimensions are compactified on a particular manifold
       The manifold is called  a Calabi--Yau manifold





caleby smooth

     
and Now