Answer:
   
   Consider the following:
   
   A cube has six sides. We live in a universe of three dimensions. Each
   dimension has two directions: front-back, right-left, up-down;
   yielding a total of six. The seventh is then the middle point, a thing
   of zero dimensions, and untouchable. Present but intangible. It
   therefore represents the holiness which is inherent in the universe.
   Thus, the physical world was created in six days, and imbued with
   sanctity on the seventh, the Shabbos. Dr. Isaac Levy includes this
   explanation in his English translation of Rabbi Samson Refa'el
   Hirsch's commentary on Numbers 16:4):
   
     The origin of this meaning is to be found in the work of the
     Creation. The visible material world created in six days received
     with the seventh day a day of remembrance of, and bond with its
     invisible L-rd and Creator, and thereby its completed consummation.
     Similarly the symbolism of the number seven in the Menora, in the
     Temple, in the Mussaf offerings, in the sprinklings of the blood on
     Yom Kippur, in the Festivals of Pesach and Succoth, in Sabbath,
     Schmita, Tumma etc. etc. The symbolism of the number eight:
     starting afresh on a higher level, an octave higher. The eighth day
     for Mila, Schmini Atzereth and Israel as the eighth of G-d's
     Creations. With the creation of Israel G-d laid the groundwork for
     a fresh, higher mankind and a fresh higher world, for that shamayim
     chadashim and the `eretz chadashah for which Israel and its mission
     is to be the beginning and instrument (Is. LXV,17).
     
     So that there are three elements in us. (a) our material sensuous
     bodies, like the rest of the created visible world = 6; (b) the
     breath of free will, invisible, coming from the Invisible One = 7;
     (c) the calling of Jew, coming from the historical choice of Israel
     = 8.
     
   Jews entered a covenant to assume a role as a "kingdom of preists".
   This preisthood requires reminding the world of the notion of "8", so
   that the world can get beyond the physical "6" and reach the
   free-willed, created, human, sanctity of "7". Eight is therefore not
   above all of creation, but beyond this universe. Eight represents
   man's ability to rise to angelic heights -- yes an image of growth,
   but not unobtainable. Man connects two worlds, eight connects those
   worlds. (Which is why the letter chet, the eighth letter, is drawn in
   the Torah as two copies of the seventh, zayin, connected by a bridge.)
   
   Which is why the laws of the covenant G-d made with Noah and thereby
   all of humanity are grouped into *seven* commandments, and the sign of
   that covenant is seen in the seven-colored rainbow.
   
   For Rav Samson Rephael Hirsch, the week gives meaning to the numbers
   six and seven. The Maharal, though, finds that the week itself is
   based on a more primary idea. He attributes the symbolism of six and
   seven to the structure of space: When you look closely you will find
   that the physical has six opposing sides, which are: top and bottom,
   right and left, front and back. All these six sides are related to the
   physical, because each side has extent, and limits physical objects.
   But, it also has in it a seventh, and this is the middle, which has no
   exposure on any side. Because it is not related to any side it is like
   the non-physical, which has no extension [takes up no volume of
   space]. (Gevuros Hashem 46)