In the parsha detailing the mitzva of Tzitzis, the Torah mentions the word Tzitzis three times over the span of two pesukim. I am wondering if any sources (from Chazal through Achronim) bring any insights as to why this is?
Maybe I am missing something and each one is needed for the simple reading, but it seems to me like there’s room for an exposition here from the fact that it’s said three times. Has anyone come across this?
The second time comes to teach us that the Tzitzis need to be made from the same material as the garment - Menachos 39b
The third time comes to teach us that placing the Tzitzis on each of the four corners is all part of one Mitzvah - not four separate Mitzvos (practical differences can be found in Menachos 37b) - Malbim
It's a way of emphasizing the commandments (the tzitzit are supposed to remind us of them). The Talmud [Nedarim 25a] says that the mitzvah of tzitzit is equivalent to all the mitzvot of the Torah. In that vein, Rashi points out that the gematria of tzitzit, 600, with 8 threads and 5 knots, is 613, the number of the commandments. [Rashi on Numbers 15:39].
Like Maurice mentions in his answer, the source dealing with this subject is tractate Nedarim 25a which says:
But isn’t it taught in a baraita: And so we found with regard to Moses our teacher. When he administered an oath to the Jewish people in the plains of Moab, that they accept the Torah upon themselves, he said to them: Know that I do not administer an oath upon you according to your understanding and the stipulations in your hearts but according to my understanding and the understanding of the Omnipresent, as it is stated: “Neither with you only do I make this covenant” (Deuteronomy 29:13). What did Moses say to Israel? Isn’t this what he said to them: Perhaps you will perform negative actions, i.e., transgressions, and say: The oath was taken according to our understanding. Due to that reason, he said to them: You take the oath according to my understanding. The Gemara clarifies: What did his warning come to exclude? Does it not serve to exclude the possibility that they give the title God, to an object of idol worship and say that this was their intention when they took an oath to worship God? The fact that Moses needed to preclude this claim indicates by inference that a person commonly takes an oath according to his own understanding. The Gemara responds: No, idol worship is also called: God, in the Bible, as it is written: “And against all the gods of Egypt I will execute judgments” (Exodus 12:12). Therefore, this would not have been a special stipulation in their minds but a misguided intention within the oath itself. Moses suspected this and therefore issued the warning. The Gemara asks: And why did Moses have to state the oath with this warning? Let him administer an oath to them with the words: That you will fulfill the mitzvot, which also includes the prohibition against idol worship. The Gemara answers: The word mitzvot, meaning commandments, could also indicate the commandments of the king, and this might be their intention if they were to take an oath in this manner. The Gemara asks: And let him administer an oath to them with the words: That you will fulfill all the mitzvot. The Gemara answers: This too does not suffice, because this phrase could indicate specifically the mitzva of ritual fringes, as the Master said: The mitzva of ritual fringes is equivalent to all the mitzvot in the Torah. Consequently, if they would accept upon themselves: All the mitzvot, they may have intended to refer only to the mitzva of ritual fringes. The Gemara asks: And let him administer an oath to them: That you fulfill the Torah. The Gemara answers: That phrase indicates only one Torah, the Written Torah and not the Oral Torah. The Gemara asks: And let him administer an oath: That you fulfill the Torahs, in the plural, to include both the Written Torah and Oral Torah. The Gemara answers: This too does not necessarily include the entire Torah, since it is possible that it indicates the Torah of the meal-offering, the Torah of the sin-offering, and the Torah of the guilt-offering. The Gemara asks: And let him administer an oath: That you fulfill the Torahs and mitzvot. The Gemara answers: This also does not include the entire Torah, because the word Torahs could indicate the Torah of the meal-offering, and mitzvot could indicate the commandments of the king. The Gemara asks: And let him administer an oath: That you fulfill the entire Torah. The Gemara answers: Fulfilling the entire Torah could indicate specifically the denial of idol worship, which is also deemed fulfilling the entire Torah, as it is taught in a baraita: Idol worship is so severe a sin that anyone who denies it is considered as though he concedes to the truth of the entire Torah. The opposite is true for someone who worships idols. Therefore, the Jewish people could have claimed that fulfilling the entire Torah denotes nothing more than not practicing idol worship. The Gemara asks: And let him administer an oath: That you fulfill the mitzva to distance oneself from idol worship and also fulfill the entire Torah. Or, alternatively, let Moses administer an oath that the Jewish people will fulfill six hundred thirteen mitzvot, so there will be no doubt as to their intention. Rather, Moses our teacher used an expression that was not troublesome for the Jews. Although he could have found another manner in which they could take an oath, and it would leave no doubt as to the correct intentions, he did not want to trouble them by employing a more complex method. Therefore, he administered the oath and stated that it was according to his understanding and the understanding of the Omnipresent.
In this context of the complete discussion mentioned in Nedarim and according to how Moshe Rabbeinu taught us in the written Torah, the first two mentions are using the term "tzitzit" (spelled ציצת) only, gematria 590, which is also the gematria of אפס זולתו like is said in the first paragraph of the Aleinu prayer. Tradition teaches that this prayer was authored by Yehoshua ben Nun, who received semicha from Moshe and filled his position leading the Jewish people after Moshe was gathered to his people.
עָלֵֽינוּ לְשַׁבֵּֽחַ לַאֲדוֹן הַכֹּל לָתֵת גְּדֻלָּה לְיוֹצֵר בְּרֵאשִׁית שֶׁלֺּא עָשָֽׂנוּ כְּגוֹיֵי הָאֲרָצוֹת וְלֺא שָׂמָֽנוּ כְּמִשְׁפְּחוֹת הָאֲדָמָה שֶׁלֺּא שָׂם חֶלְקֵֽנוּ כָּהֶם וְגוֹרָלֵֽנוּ כְּכָל הֲמוֹנָם: שֶׁהֵם מִשְׁתַּחֲוִים לָהֶֽבֶל וָרִיק וּמִתְפַּלְּ֒לִים אֶל אֵל לֹא יוֹשִֽׁיעַ, וַאֲנַֽחְנוּ כּוֹרְ֒עִים וּמִשְׁתַּחֲוִים וּמוֹדִים לִפְנֵי מֶֽלֶךְ מַלְכֵי הַמְּ֒לָכִים הַקָּדוֹשׁ בָּרוּךְ הוּא, שֶׁהוּא נוֹטֶה שָׁמַֽיִם וְיוֹסֵד אָֽרֶץ, וּמוֹשַׁב יְקָרוֹ בַּשָּׁמַֽיִם מִמַּֽעַל, וּשְׁ֒כִינַת עֻזּוֹ בְּגָבְ֒הֵי מְרוֹמִים, הוּא אֱלֺהֵֽינוּ אֵין עוֹד, אֱמֶת מַלְכֵּֽנוּ אֶֽפֶס זוּלָתוֹ כַּכָּתוּב בְּתוֹרָתוֹ וְיָדַעְתָּ הַיּוֹם וַהֲשֵׁבֹתָ אֶל לְבָבֶֽךָ כִּי יְהֹוָה הוּא הָאֱלֺהִים בַּשָּׁמַֽיִם מִמַּֽעַל וְעַל הָאָֽרֶץ מִתָּֽחַת אֵין עוֹד:
Like it says there, this emphasizes that G-d, our King is true (that G-d's seal and the seal of the Torah is Truth אמת). There is no other King beside Him. These two expressions pertain to the tzitzit of white and techelet (blue) being two separate and distinct categories of this single mitzvah.
The third mention of tzitzit in the paragraph is "l'tzitzit" (לציצת) which is gematria 620, not 613 like Rashi mentions in connection to B'Midbar 15:39. To support his proposed gematria count of all the commandments of the Torah, Rashi adds a letter Yud to the spelling found in the written Torah.
In this context, it should be noted that if one adds a letter or omits a letter from the written Torah, it makes the whole Torah invalid.
Like the text of the written Torah emphasizes, the word לציצת is emphasizing "all the mitzvot of G-d", (not only the 613 mitzvot of the B'nai Yisroel) are one.
וְהָיָה לָכֶם לְצִיצִת וּרְאִיתֶם֯ אֹ֯תוֹ וּזְכַרְתֶּם֯ אֶ֯ת־כָּל־מִצְוֹת֯ יְ֯הֹוָה
That לציצת, when including its five letters, is gematria 625 (הכתר או כתר ה׳) which is also the gematria of כל מצות יהוה when including its three words and ten letters.
This means both the 613 mitzvot of B'nai Israel and the 7 Mitzvot of B'nai Noach (like is emphasized in Nedarim quoting Moshe Rabbeinu, "Neither with you only do I make this covenant..."), all of which were commanded by G-d directly, are essential and necessary for revealing the one, true G-d's Kingship and are complementary to each other. That 620 is gematria כתר (Crown) alluding to G-d's Kingship over everyone.
That this 620 refers to the 613 mitzvot of B'nai Yisrael and the 7 mitzvot of the B'nai Noach follows the explanation of Rabbeinu Nissim ben Reuven of Girona to this expression in Nedarim. He explains that this is also the plain meaning from D'varim 29:13-14 which says:
וְלֹ֥א אִתְּכֶ֖ם לְבַדְּכֶ֑ם אָנֹכִ֗י כֹּרֵת֙ אֶת־הַבְּרִ֣ית הַזֹּ֔את וְאֶת־הָאָלָ֖ה הַזֹּֽאת׃ כִּי֩ אֶת־אֲשֶׁ֨ר יֶשְׁנ֜וֹ פֹּ֗ה עִמָּ֙נוּ֙ עֹמֵ֣ד הַיּ֔וֹם לִפְנֵ֖י יְהֹוָ֣ה אֱלֹהֵ֑ינוּ וְאֵ֨ת אֲשֶׁ֥ר אֵינֶ֛נּוּ פֹּ֖ה עִמָּ֥נוּ הַיּֽוֹם׃
That Rabbeinu Nissim says this is referring to the non-Jewish nations, who were not standing at Har Sinai together with Israel and receiving the Torah. In fact, like is recorded in the Talmud, they had refused it. Their obligation to the Creator of us all is to His commandments which He gave to them previously with Adam HaRishon and Noach. That their obligation to keep the seven mitzvot of the children of Noach was not nullified or replaced, but renewed and re-emphasized at the time of the giving of the Torah.
I would share what Rabbenu Bachaya says on this matter, in Parshat Shlach, Bam.15:38 It's a quite long comment but definitely worth reading and studying. I can't copy and paste his whole comment because it exceeds the number of characters we can possibly type in. Here is part of his commet with regards to your question about why three times the word Tzitzit is used in Bamidbar chapter 15:
"[...]The reason that we make five knots when attaching the tzitzit to the talit is that we have been taught that the fulfillment of this commandment is equivalent to all the 613 commandments of the Torah. The word ציצית has a numerical value of 600. When you add the eight threads and the five knots you have the number 613, an allusion to all the commandments of the Torah. This is the mystical dimension of (15,39) “when you see it (the fringe) you will remember all the commandments of the Torah and remember to perform them, etc. Do not object that the word ציצת appears three times in this passage spelled without the letter י before the letter ת, so that the numerical value of the word is only 590 and not 600. The letter ל in the word לציצת in verse 38 makes up for the missing thirty through the absence of three times the letter י, so that the combined numerical value of each word ציצת, לציצת, ציצת is 1800, which divided by three still amounts to 600. The reason the word has been mentioned three times in our paragraph is to serve as an allusion that the statement that a single commandment is equivalent to all the 613 commandments is applicable to a total of three such commandments, i.e. he commandment of tzitzit, the commandment of observing the Sabbath, and the commandment not to engage in idolatry (Compare Chulin 5) [...]"
