kiddush - Why is "Vayechulu" said 3 times on Leil Shabbat?

I'm curious about the specific number 3 here. Is there a reason or "meaning" (such as from Zohar, etc.) as to why Vayechulu (Breishit 2:1-3) is said 3 times on Friday night (a non-Yom-Tov one)?

• In the Amidah


• After the Amidah prior to Magen Avot



• As part of Kiddush at home

Answer

This is a very old custom, which is cited in the Sefer Rokeach Hilkhot Shabat 49 by Eleazar of Worms.¹ The meaning of the three is given by the Kaf haChayim in his commentary to Orach Chayim 268:34, saying that it refers to the three worlds (see here), the higher, the middle and the lower one:

ובצרור המור ט"א על ג"פ נגד ג' עולמות עולם עליון ואמצעי ותחתון לומר שהקב"ה ברא כולם.‏

This idea is already mentioned by R' Avraham Saba, who says in his commentary, Tzeror haMor² that these three worlds are alluded in the word yom that occurs three times in Vaykhulu:³

וַיְכֻלּ֛וּ הַשָּׁמַ֥יִם וְהָאָ֖רֶץ וְכָל־צְבָאָֽם׃ וַיְכַ֤ל אֱלֹהִים֙ בַּיּ֣וֹם הַשְּׁבִיעִ֔י מְלַאכְתּ֖וֹ אֲשֶׁ֣ר עָשָׂ֑ה וַיִּשְׁבֹּת֙ בַּיּ֣וֹם הַשְּׁבִיעִ֔י מִכָּל־מְלַאכְתּ֖וֹ אֲשֶׁ֥ר עָשָֽׂה׃ וַיְבָ֤רֶךְ אֱלֹהִים֙ אֶת־י֣וֹם הַשְּׁבִיעִ֔י וַיְקַדֵּ֖שׁ אֹת֑וֹ כִּ֣י ב֤וֹ שָׁבַת֙ מִכָּל־מְלַאכְתּ֔וֹ אֲשֶׁר־בָּרָ֥א אֱלֹהִ֖ים לַֽעֲשֽׂוֹת׃

These worlds are relevant, because by reciting these verses we're partners with Hashem in the creation of the world according to Shabbat 119b:

דאמר רב המנונא, כל המתפלל בערב שבת ואומר ויכולו מעלה עליו הכתוב כאילו נעשה שותף להקדוש ברוך הוא במעשה בראשית, שנאמר ויכולו אל תקרי ויכולו אלא ויכלו.‏

^{1 The source is cited in the Mishnah Berurah (45) to Orach Chayim 271:10
2 See penultimate row on page
3 I've found this source in the Beurei haTefilah newsletter}

Kaf haChayim also gives another explanation citing another midrash. We read three times asher in Vaykhulu (also in bold above), the same number as it occurs in the verse about the red heifer (Numbers 19:2):

זֹ֚את חֻקַּ֣ת הַתּוֹרָ֔ה אֲשֶׁר־צִוָּ֥ה יְהוָ֖ה לֵאמֹ֑ר דַּבֵּ֣ר ׀ אֶל־בְּנֵ֣י יִשְׂרָאֵ֗ל וְיִקְח֣וּ אֵלֶיךָ֩ פָרָ֨ה אֲדֻמָּ֜ה תְּמִימָ֗ה אֲשֶׁ֤ר אֵֽין־בָּהּ֙ מ֔וּם אֲשֶׁ֛ר לֹֽא־עָלָ֥ה עָלֶ֖יהָ עֹֽל׃

This is the statute of the law which the Hashem hath commanded, saying: Speak unto the children of Israel, that they bring thee a red heifer, faultless, wherein is no blemish, and upon which never came yoke.

Who says these verses atones for the sins commited, just as the red heifer purifies the uncleanliness:

ובמדרש אמר ג"פ אשר בויכלו כמו בפ' פרה אדמה להורות שהאומר ג"פ ויכלו מתכפר.‏