Shemini Atzeret, the "Eighth Day of Assembly", is a biblical holiday which immediately follows the 7-day festival of Sukkot. It always falls on the 22nd of Tishrei as a one-day holiday in Israel, and it is observed on both the 22nd and 23rd of Tishrei outside of Israel, in the diaspora.
The custom of celebrating 2-days of the holiday outside of Israel is based on historical confusion around the exact start of the new month. By maintaining two-days, at least one day of the holiday is on the correct day established by the Torah. The custom has been maintained following the creation of a fixed calendar and continues today.
Although Shemini Atzeret is connected to the 7-day holiday of Sukkot by reference to the 8th Day, it was established in the Torah as a distinct holiday on which we recite the She' hechiyanu blessing and are not obligated to sit in the Sukkah. As a separate, yet connected, holiday, it hints to the dualism also found between the physical and spiritual realms.
The sages suggest that this "Eighth Day" of Shemini Atzeret, is one final day after the holiday of Sukkot in which G-d's people can remain together before He ascends back to the higher realms. While Sukkot draws down the surrounding Light of Mercy (or makif) for all 70 nations of the world who dwell in the Sukkah, the Talmud states that only one bull was offered on the holiday of Shemini Atzeret hinting to the unity achieved during the holiday of Sukkot. The root of the word Atzeret is "atzor" which means to stop or tarry, further suggesting to us to stop, linger and remain in the Sukkah for one more day - the 8th day.
Simchat Torah, "Rejoicing with the Torah", is a rabbinical creation and holiday of joy which commemorates the completion of the annual Torah cycle, followed immediately by the commencement of the next cycle.
The specific practice of reading the end of Deuteronomy and the beginning of Genesis, which has become known as Simchat Torah, is conducted on the holiday of Shemini Atzeret and the two names are used inter-changeably in Israel. In the diaspora however, the Simchat Torah practices and observances are conducted on the second day of Shemini Atzeret, a day considered by many Jews as the distinct Rabbinical holiday of Simchat Torah.
Hakafot - On Simchat Torah, we celebrate by taking out all of the Torah scrolls and circling the Bimah in the synagogue seven times, or making seven circuits (Hakafot). Similar to how a bride circles the groom 7 times in an orthodox marriage ceremony, this day similarly represents a wedding day - our marriage to G-d where the congregation are the bride circling G-d and the Torah, the Groom. The Zohar states that we are able to draw down the divine energy on the day and draw light into our new vessel created during the prior holidays of Rosh Hashanah, Yom Kippur and Sukkot. It is the only holiday where we bring out the Torahs in the evening while we make the Hakafot on erev Simchat Torah.
Ending and Beginning the Cycle - After we complete the Hakafot prior to the morning services, we read the final Torah portion, Ve'zot Habracha (the only Torah portion not read on Shabbat). The reading is a very joyous time and everyone in the community - including children, are called to the Torah. We read the final portion, which is immediately followed by reading the first part of the first portion of the Torah, Bereshit. We connect to the continuity found in the completion of one cycle and the seamless new beginning of another annual Torah reading. The final reading from the Book of Deuteronomy is conducted by the Hatan Torah and the first reading from Genesis is reserved for the Hatan Bereshit - the person that begins the new Torah cycle for the community.
Prayer for Rain - During the Musaf prayer on Shemini Atzeret, we offer a special prayer for rain. Rain is considered a source of growth and abundance, helping to support and "water" the clean and perfected souls we achieved through our acts of Teshuva during the holidays. We likewise connect to the "seed" of the Torah with our connection to Bershit which likewise needs watering. The rain helps us maintain our connection made during the holidays and preserve it through the New Year and new Torah cycle.
Shemini, the 8th day, connects us to the energy of miracles. Eight is one above 7, the natural sequence found in 7 days in a week, 7 musical notes and 7 colors of the spectrum - 8 is a day beyond the natural world. An eight, when viewed on its side, connects us to the infinite.
Following the days of judgement at Rosh Hashanah & Yom Kippur and the 7-days of surrounding Light and mercy provided through Sukkot, we can use this 8th day of Shemini Atzeret and Simchas Torah to manifest the connection we've been working on throughout the holidays. We connect with the Creator in the highest possible physical manner with the simple act of dancing and "Rejoicing with the Torah."
Chasidic masters suggest that on Simchat Torah our feet have a chance to transport the head (represented by the Torah) wherever it wants to go - an act of deference to the will of G-d.
The Zohar has a beautiful discussion of the holiday in Parsha Pinchas. It reveals how the judgement of Rosh Hashanah and Yom Kippur and the mercy of Sukkot is balanced by Shemini Atzeret and Simchat Torah. The Zohar says in part, "... On the day of Shemini Atzeret, which is Simchat Torah, there is the mating of the body, namely the Central Column, called ‘body,’ taking place. This is the union of all parts, for it includes the mating of the Left side, of Rosh Hashanah and Yom Kippur as well as the union of the Right side—the holiday of Sukkot, since the Central Column incorporates the Right and the Left. Thus all is one, and this is overall perfection."
Once we've cancelled our vows, sought and provided forgiveness and attempted to clear the slate for the New Year during Rosh Hashanah and Yom Kippur, we receive the surrounding Light of mercy during Sukkot. We've done the work to purify ourselves to receive the beneficent Light of the Creator. We can now use the eighth day of Shemini Atzeret, a day beyond the natural world where miracles exist, to connect to the Light. We make our physical connection by dancing and singing with the Torah and our spiritual connection by reading from Bereshit, the seed level of creation which starts the annual Torah cycle.
Use these tools and deeper understanding about the holiday to have a powerful Shemini Atzeret & Simcha Torah connection!