The Gra (Aderes Eliyahu) in the beginning of Ha'azinu has many dikdukim in the possuk, האזינו השמים ואדברה ותשמע הארץ אמרי פי. One of the points he raises it why the heavens is only afforded three words, האזינו השמים ואדברה while the earth gets four, ותשמע הארץ אמרי פי? He explains that שמים refers to the Torah שבכתב and the ארץ refers to Torah שבעל פה. He says the 4 and the 3 parallel the two types of שין on the tefillin of the head, the right side has a four pronged שין and the left side has a three pronged. These also correspond to the three ברכת השחר in the morning and 4 in the evening. How do these ideas all come together, what does the written Torah have to do with the number 4 and the oral Torah with the number 3?
The Gemarah Shabbas (104a) says ג,ד stands for גומל דלים. In the terminology of Chassidus ג refers to the middah of yesod which is the giver and the letter ד refers to malchut, the recipient. The written Torah is the giver, it is given to Klal Yisrael from Hashem. That is why it matches up with the number 3, the giver. The oral Torah is the Torah composed by the recipients, Klal Yisrael. The 4 pronged שין is on the right side of the tefillin, the side of chesed, the masculine side, the giver. The 3 pronged שין in on the left side, that of gevurot, the feminine side, the recipient. That is why there are 3 avos, three givers and 4 imahot, 4 receivers. The morning is the time of chesed as well, there is a possibility of a bright new day. The evening is the time of gevurot, of the time of taking in the events of the day. That is the matchup of 3 to the written Torah, שמים and 4 to the oral Torah, ארץ.
The name of the Alter Rebbe is שניאור which according to the Maharshal refers to שני אור, two forms of light. The avodah of chassidus is to take the ארץ and align it with שמים. When the recipient is willing to acknowledge the giver that is the ultimate tikkun. This is is hinted to by the fact that the original 19-20 Kislev liberation happened on a Tuesday- Wednesday, יום ג ויום ד. The day is to celebrate the spreading forth of the recognition of the complete presence of the שמים, the נותן in the ארץ.
The Bas Ayin notes the the fact that Kislev is three months after Tishray and parallels this to the story of Yehuda and Tamar. Rav Levi Yitzchak notes that there are two forms of three which correspond with two forms of giving the Torah. The Torah was given in Sivan, the third month from the count from Nissan. The Gemarah Shabbas (88a and Rav Nissan Gaon ad loc.) note many forms of three that were present at the giving of the Torah. Moshe was the third child from the third shevet, Levi. This is present at the giving of the written Torah from Hashem, the אתערותא דלעילא which happens in the third month from the method of counting from Nissan, the month of the greatest אתערותא דלעילא. Chanukah is the holiday which is the precursor to the oral Torah and hence it happens in the third month counting from Tishray. In other words, the count from אתערותא דלתתא, from man's own actions. That is why Chanukah happened only through mesiras nefesh, it had to happen through the actions of man. And the miracle happened through the חשמונאים, the descendants of Levi, the third tribe. It is the oral Torah that leads us through the golus. It is what distinguishes us from the gentiles. While they may have the Septuagint, access to the written Torah, they do not have access to the oral Torah. This is the three that leads to four, when the recipients, Klal Yisrael play an active role, and it is that fusion that guides us through the centuries.