The Paradox Of The Eighth Day
There is a statement in the Zohar  that the three pilgrimage festivals correspond to the three Patriarchs.
Pesach corresponds to Avraham who gave the command:  "Knead [a dough] and make cakes [of matzah]."
Shavuos corresponds to Yitzchak, for Shavuos commemorates the Giving of the Torah, associated with the call of the shofar,  which shares a connection with the ram of Yitzchak. 
And Sukkos corresponds to Yaakov, of whom it is said,  "And he made sukkos ('booths') for his herds." This is the first time the word sukkos is mentioned in the Torah.
Shemini Atzeres corresponds to Yosef,  "the posterity of Yaakov," just as Shemini Atzeres is a continuation of the holiday of Sukkos.
With regard to Shemini Atzeres, we find two contradictory factors.
On one hand it is considered as a festival in its own right.  As such, we recite the blessing Shehechiyanu on Shemini Atzeres, in contrast to the seventh day of Pesach, when this blessing is not recited. For the seventh day of Pesach is not a separate holiday. And yet, its very name Shemini Atzeres, lit. "the assembly of the eighth day," indicates that it is a continuation of the holiday of Sukkos.
The explanation is that the connection between Shemini Atzeres and Sukkos parallels that between Yosef and Yaakov.
Yosef is an extension of Yaakov, and his contribution represents transmitting Yaakov's influence into the lower realms.
Nevertheless, the ability to transmit this influence represents a distinct advantage which Yaakov himself does not possess.
Similarly with regard to Shemini Atzeres, this holiday internalizes all the influence of the holiday of Sukkos (which in turn includes within it the influence of Pesach and Shavuos, as the attributes of Yaakov includes the attributes of Avraham and Yitzchak).
The holiday of Sukkos represents an encompassing light (or makkif, in Chassidic terminology), just as a sukkah encompasses the people who dwell within. Shemini Atzeres makes it possible to internalize these influences.
This is implied by the name Atzeres which means "gathering in," collecting and concentrating the influence of Sukkos. 
Nevertheless, the fact that Shemini Atzeres makes it possible for these influences to be internalized indicates that it possesses a more powerful potential. And this more powerful potential enables the revelations which transcend the world - and can only be drawn into the world in an encompassing manner - to be internalized within the framework of worldly existence.
Moreover, there is another significant distinction between Shemini Atzeres and Sukkos.
On Sukkos, seventy bulls would be sacrificed, corresponding to the seventy nations of the world.  Since the influence of the holiday of Sukkos is drawn down in an encompassing manner, it is possible for the gentile nations to derive nurture from it. And therefore, the seventy bulls were sacrificed to refine the seventy nations.
On Shemini Atzeres, by contrast, only one bull and one ram was offered, pointing to the singular bond between G-d and the Jewish people: "This is for you alone, and not for any strangers with you."  Although the influence which was revealed before is drawn down into this world, it is drawn down in such a manner that no "strangers" can receive any nurture from it.
A parallel to this can be seen in the Divine service of Yosef, who despite his involvement in Egyptian society, "the nakedness of the land," was able to cling to G-d in a perfect bond.
The Significance Of The Number Eight
There is another dimension which reflects the connection between Yosef and the holiday of Shemini Atzeres.
One of the reasons the holiday is given the name Shemini Atzeres although it is "a festival in its own right" is to emphasize the significance of the number eight.
For eight reflects a light that transcends the limits of the natural order, including the limits of seder hahishtalshelus, the chainlike progression of spiritual worlds.  The natural order is structured in sets of seven, and the number eight reflects a light that transcends these limits.
This level is revealed on Shemini Atzeres.
And because Shemini Atzeres is associated with a transcendent light that is not confined by any limits whatsoever, it can internalize the spiritual influences which are so lofty that they can be revealed on the holiday of Sukkos only in an encompassing manner.
We find a similar concept with regard to the mitzvah of circumcision which is carried out on the eighth day of a child's life.
The mitzvah of circumcision activates the level associated with the number eight. This in turn makes it possible to draw down the highest levels of holiness, those which transcend the world (in Kabbalistic terminology, the level of Kesser  which transcends seder hahishtalshelus, and draw them down and connect them with the limb which is described as "yesod, the end of the torso." 
Through the mitzvah of circumcision, our physical flesh is refined and brought into an eternal bond with G-d.
The level of eight which shines on Shemini Atzeres is associated with Yosef  who is described as "the tzaddik, the yesod of the world." 
- (Back to text) Vol. III, p. 257b. See also Menoras HaMeor (from R. Yonah Alnakava), Vol. II, Hilchos Rosh Chodesh in the name of
R. Yehudah, the son of Rabbeinu Asher.
- (Back to text) Bereishis 18:6.
- (Back to text) Shmos 19:19.
- (Back to text) Pirkei d'Rabbi Eliezer, ch. 31, cited by Rashi, Shmos 19:13; Midrash Aggadah, Bereishis 22:13. See also the
maamar Shuvah, 5641, in the series of maamarim entitled
- (Back to text) Bereishis 33:17.
- (Back to text) See Zohar, Vol. I, p. 208b.
- (Back to text) Sukkah 48a.
- (Back to text) On this basis, we can appreciate the implications of
the wording used by Rashi, Sukkos, op. cit.: "[Shemini
Atzeres is] a holiday in its own right, for we do not
dwell in sukkos." On the surface, the fact that we do not
dwell in sukkos is merely a negative factor, indicating
that it is not part of the holiday of Sukkos; it does not
show that it is a holiday of its own. Based on the above,
however, it can be explained that the intent is that we
no longer dwell in Sukkos, but that the encompassing
influence of Sukkos is internalized within our homes.
- (Back to text) Sukkah 55b.
- (Back to text) Cf. Mishlei 5:17. The connection of this verse to Shemini Atzeres is explained in the maamarim entitled, BaYom
HaShemini Atzeres from 5709 and 5710.
- (Back to text) See Likkutei Torah, Devarim 87d. See also the sichah in this series for Chanukah and the sources mentioned there.
- (Back to text) This is reflected in the fact that the mitzvah of
circumcision is one of the two positive commandments for
which the lack of observance is punishable by kareis,
"the soul being cut off." Kareis shares the same Hebrew
letters as Kesser.
With the exception of the Paschal sacrifice (which is
connected to circumcision as cited in note 30), there
are no other positive mitzvos for which the lack of
observance is punishable by kareis. The rationale is
that generally, the level of Kesser is too lofty to be
drawn down through positive activity. Instead, it is
holding back, refraining from activity, which draws down
The uniqueness of the mitzvos of circumcision and the
Paschal sacrifice is that they make it possible for the
light of Kesser to be drawn down through positive
activity, and thus to be enlothed within vessels.
- (Back to text) Hakdamas Tikkunei Zohar 17a, Siddur Tehilas HaShem, p. 125.
- (Back to text) Cf. Mishlei 10:25.