The number three and the festivals
Why three of them and what do they all represent?
This article in part appears on my website - livingwithmitzvos.com. I have also authored a book entitled Megillas Esther Explained(Mosaica Press) that is due to soon come out. I have also several articles under Collections - Megillas Esther Explained.
There are the three main festivals in the Jewish Year of Pesach, Shavuos and Sukkos. Since these are so central they parallel the three Avos.

(א)הלכות ראש חודש
... ושמעתי מאחי הר"י טעם לדבר לפי שהמועדים נתקנו כנגד אבות פסח כנגד אברהם דכתיב (בראשית יח) לושי ועשי עוגות ופסח היה שבועות כנגד יצחק שתקיעת שופר של מתן תורה היה בשופר מאילו של יצחק סוכות כנגד יעקב דכתיב (בראשית לד) ולמקנהו עשה סוכות.

This translates as:
And I heard from my brother, the Ri, a reason for this matter: because the Festivals were established corresponding to the Patriarchs. Pesach corresponds to Abraham, as it is written, "Knead it and make cakes" (Genesis 18:6), and it was [the time of] Pesach. Shavuot corresponds to Isaac, because the blowing of the Shofar at the giving of the Torah was from the ram of Isaac. Sukkot corresponds to Jacob, as it is written, "And for his livestock he made booths [Sukkot]" (Genesis 33:17).
Since these are so central, the parallel the three main pillars of emuna, faith. (In Sefer HaIkkarim (“Book of Principles”), the 15th-century philosopher Rabbi Joseph Albo narrows the core foundations of Jewish belief down to three primary fundamentals (Ikkarim Kollelim). While Maimonides (the Rambam) famously outlined Thirteen Principles of Faith, Rabbi Albo argues that every divine law relies essentially on these three “roots,” and that rejecting any one of them undermines the entirety of the Torah,)

הדרך הנכון שיראה לי בספירת העקרים שהם שרשים ויסודות לתורה האלהית הוא, כי העקרים הכוללים וההכרחיים לדת האלהית הם שלשה, והם מציאות השם, וההשגחה לשכר ולעונש, ותורה מן השמים.

It seems to me that the general and essential principles of divine law are three: existence of God, providence in reward and punishment, and divine revelation.

In Kabbalah, the significance of the number three is that it represents the two extremes of right and left, that then unite in the middle. This is seen in the three Avos.
Avrohom as the first one represents chesed, kindness, that spreads outwards and this is seen in the right side. This parallels Pesach that is the time when Hashem did kindness to the Jews as seen in the Exodus.
Yitzchok as the second, comes to limit this chesed so that it is placed in the right parameters. This is seen in the left side of Din, judgement, that puts boundaries in place to ensure the right balance. This is paralleled in the Torah given on Shavuos that comes with the mitzvos that are split into positive ones (chesed) together with negative ones (din) that guides the Jews of what yes and not to do.
Yakov as the third one that combines them is known as בחיר האבות, the choicest of all three since he represents Tiferes that sees a harmony of how these all come together. This is seen in the festival of Sukkos where all Jews come together into the Sukka.

וַחֲכָמִים אוֹמְרִים: אַף עַל פִּי שֶׁאָמְרוּ אֵין אָדָם יוֹצֵא יְדֵי חוֹבָתוֹ בְּיוֹם טוֹב הָרִאשׁוֹן בְּלוּלָבוֹ שֶׁל חֲבֵירוֹ, אֲבָל יוֹצֵא יְדֵי חוֹבָתוֹ בְּסוּכָּתוֹ שֶׁל חֲבֵירוֹ, דִּכְתִיב: ״כׇּל הָאֶזְרָח בְּיִשְׂרָאֵל יֵשְׁבוּ בַּסּוּכּוֹת״, מְלַמֵּד שֶׁכׇּל יִשְׂרָאֵל רְאוּיִם לֵישֵׁב בְּסוּכָּה אַחַת.

And the Rabbis say: Although they said that a person does not fulfill his obligation on the first day of the Festival with the lulav of another, he fulfills his obligation with the sukka of another, as it is written: “All the homeborn in Israel shall reside in sukkot” (Leviticus 23:42). This teaches that all of the Jewish people are fit to reside in one sukka. If the value of one sukka were divided among all the Jewish people, no individual would have a peruta stake in it; therefore, no individual could be considered even a part-owner of the sukka. The only way the entire Jewish people could fulfill the mitzva in one sukka is by residing in a communal sukka that does not belong to any of them. Apparently, there is no obligation to reside specifically in one’s own sukka.

This idea of unity is also seen in another aspect of Sukkos, seen in the taking of the Arba Minim:

(יב) דָּבָר אַחֵר, פְּרִי עֵץ הָדָר, אֵלּוּ יִשְׂרָאֵל, מָה אֶתְרוֹג זֶה יֵשׁ בּוֹ טַעַם וְיֵשׁ בּוֹ רֵיחַ, כָּךְ יִשְׂרָאֵל יֵשׁ בָּהֶם בְּנֵי אָדָם שֶׁיֵּשׁ בָּהֶם תּוֹרָה וְיֵשׁ בָּהֶם מַעֲשִׂים טוֹבִים. כַּפֹּת תְּמָרִים, אֵלּוּ יִשְׂרָאֵל, מָה הַתְּמָרָה הַזּוֹ יֵשׁ בּוֹ טַעַם וְאֵין בּוֹ רֵיחַ, כָּךְ הֵם יִשְׂרָאֵל יֵשׁ בָּהֶם שֶׁיֵּשׁ בָּהֶם תּוֹרָה וְאֵין בָּהֶם מַעֲשִׂים טוֹבִים. וַעֲנַף עֵץ עָבֹת, אֵלּוּ יִשְׂרָאֵל, מָה הֲדַס יֵשׁ בּוֹ רֵיחַ וְאֵין בּוֹ טַעַם, כָּךְ יִשְׂרָאֵל יֵשׁ בָּהֶם שֶׁיֵּשׁ בָּהֶם מַעֲשִׂים טוֹבִים וְאֵין בָּהֶם תּוֹרָה. וְעַרְבֵי נָחַל, אֵלּוּ יִשְׂרָאֵל, מָה עֲרָבָה זוֹ אֵין בָּהּ טַעַם וְאֵין בָּהּ רֵיחַ, כָּךְ הֵם יִשְׂרָאֵל יֵשׁ בָּהֶם בְּנֵי אָדָם שֶׁאֵין בָּהֶם לֹא תּוֹרָה וְלֹא מַעֲשִׂים טוֹבִים, וּמָה הַקָּדוֹשׁ בָּרוּךְ הוּא עוֹשֶׂה לָהֶם, לְאַבְּדָן אִי אֶפְשָׁר, אֶלָּא אָמַר הַקָּדוֹשׁ בָּרוּךְ הוּא יֻקְשְׁרוּ כֻלָּם אֲגֻדָּה אַחַת וְהֵן מְכַפְּרִין אֵלּוּ עַל אֵלּוּ, וְאִם עֲשִׂיתֶם כָּךְ אוֹתָהּ שָׁעָה אֲנִי מִתְעַלֶּה, הֲדָא הוּא דִכְתִיב (עמוס ט, ו): הַבּוֹנֶה בַשָּׁמַיִם מַעֲלוֹתָו, וְאֵימָתַי הוּא מִתְעֲלֶה כְּשֶׁהֵן עֲשׂוּיִין אֲגֻדָּה אַחַת, שֶׁנֶּאֱמַר (עמוס ט, ו): וַאֲגֻדָּתוֹ עַל אֶרֶץ יְסָדָהּ, לְפִיכָךְ משֶׁה מַזְהִיר לְיִשְׂרָאֵל: וּלְקַחְתֶּם לָכֶם בַּיּוֹם הָרִאשׁוֹן.

(12) Another matter: “The fruit of a pleasant [hadar] tree” – this is Israel; just as the citron has taste and has fragrance, so Israel has people among them who have Torah and have good deeds. “Branches of date palms” – this is Israel; just as the date palm has taste but has no fragrance, so Israel has people among them who have Torah but do not have good deeds. “A bough of a leafy tree” – this is Israel; just as the myrtle has fragrance but has no taste, so Israel has people among them who have good deeds but do not have Torah. “Willows of the brook” – this is Israel; just as the willow has no taste and has no fragrance, so Israel has people among them who do not have Torah and do not have good deeds. What does the Holy One blessed be He do to them? To eradicate them is not possible. Rather, the Holy One blessed be He said: Let them all be bound together in a single bundle and they will atone for one another. If you do so, at that moment, I will be ascendant [mitaleh]. That is what is written: “Who built His upper chamber [maalotav] in the heavens” (Amos 9:6). When is He ascendant? When they are made into a single group, as it is stated: “And established His group upon earth” (Amos 9:6). Therefore, Moses exhorts Israel and says to them: “You shall take for you on the first day.”

As Sukkos is approaching, let us pray for unity amongst all the different type of Jews and hope for the speedy coming of the Moshiach in our days - Amen.
This article in part appears on my website - livingwithmitzvos.com. I have also authored a book entitled Megillas Esther Explained (Mosaica Press) that is due to soon come out. I have also several articles under Collections - Megillas Esther Explained.