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The Mitzvah of Techeiles: 
As stated in the verse, the Biblical Mitzvah of Tzitzis requires that the Tzitzis strings contain both a set of white strings and a set of Techeiles strings. These two colors of string are all included within the same single Mitzvah of Tzitzis, and hence only a single positive command is fulfilled when one wears Techeiles. These two colors are independent of each other, and thus even if one were to wear only Techeiles strings without white strings, or only white strings without Techieles strings, he has still fulfilled the positive command. However, some Poskim hold that if one made all the strings Techeiles, without any white strings, the Tzitzis is invalid.
How many strings are to be of Techeiles and how many of Lavan? In times that Techeiles was available the Mitzvah of Tzitzis was to take two strings which are dyed with the color called Techeiles and two non-dyed strings which are called the white strings [Chutei Lavan].
Does Techeiles exist today and is one to be scrupulous to use such strings for his Tzitzis?
Techeiles has been extinct for many generations, since the times of the Rabanan Savuraiy which lived after the period of the Amoraim. It was mystically hidden from the Jewish people due to the exile of the Shechina. In the year 1889, the Admur of Radzin [Rav Gershon Henich of Radzin] wrote a Halachic responsa acclaiming to have found the identity of the Techeiles carrying fish and that it once again should be used as part of the Tzitzis strings, as is the Biblical requirement. In light of this discovery, many Jews began using the Techeiles strings and more than 12000 Jews from various countries reinstated this Mitzvah. Practically, although some Poskim defended the reinstation of the Techeiles, majority of Poskim vehemently opposed it, including the Rebbe Rashab and the Rebbe and so is the Chabad custom as well as the custom of majority of Jewry. So is also the ruling according to the Arizal which states that the Mitzvah of Techeiles will only be reinstated when Moshiach comes.
The purpose of Techeiles and its effect even today:
The Techeiles of the Tzitzis represents the level of Makifim, which blind the eyes of the Kelipa and draws fear of Heaven. It is the most powerful form of Makifim. This is because this Makif comes from the level of Malchus and descends to the worlds of Biyah to prevent the nurture of the Kelipos from the G-dliness found in those levels. The spiritual effect of Techeiles assists a person in fulfilling the verse of “Lo Sasuru”, not to stray after one’s heart or eyes.
The ray of the Techeiles in today’s times: Although we have lost the ability to perform the Mitzvah of Techeiles due to the exile, nevertheless the spiritual ray of Techeiles still shines even today.
 Aruch Hashulchan 9/2-3
 This follows the opinion of Rashi. [Aruch Hashulchan ibid]
 Rambam beginning of Tzitzis
 Admur 11/1; Tur 11; Rashi and Tosafus Menachos 38a [brought in Beis Yosef ibid]; See Semag 27; Aruch Hashulchan 9/2
Other Poskim: Some Poskim rule that only one out of the 8 strings is dyed in blue. Thus only half of one of the 4 strings was dyed. [Rambam Tzitzis 1/6] Other Poskim rule that two of the 8 strings are dyed with Techeiles, and hence two half strings [or one whole string] is dyed. [Raavad ibid]
 The reason these strings were white: The Mitzvah of Techeiles was only in affect for two of the four strings. Consequently, the other two strings are not dyed in any color and hence remain in their natural white state. [Admur ibid]
 See Avoda Zara 39a that Techeiles was still on the market in times of the Amoraim
 Arizal in Peri Eitz Chaim Shaar Hatzitzis 5, brought in Igros Kodesh [of Rebbe Rashab] 1/158
 See his Sefer Eiyn Hatecheiles
 See Hisorerus Teshuvah 2/280; Orchos Rabbeinu 1/246 in name of Maharsham
 Aruch Hashulchan 9/4; Yeshuos Malko 1-3; Or Layesharim ; Sheilas Shalom 64; Rav Elyashiv in Koveitz Teshuvos 2; Teshuvos Vehanhagos 1/29; 4/5; Or Letziyon 2/44-13
 Igros Kodesh [of Rebbe Rashab] 1/158 in response to the authors son [Rav Mordechai Yosef of Radzhin] “I thank you for sending me the Sefer of your father, although regarding the actual matter I cannot consent at all.”
 Igros Kodesh 17/232 [printed in Shulchan Menachem 1/45]
 “We only have the words of the Rebbe Rashab” [Rebbe ibid]
 The reasons: Many reasons are attributed for why one should not wear Techeiles today: 1) Many opinions exist regarding the identity of the true Techeiles carrying fish, and hence one should not enter himself into doubt. The reason for this is because Lechatchila the strings of the Tzitzis are to be of the same color as the garment, and hence if the Techeiles color is incorrect, one is transgressing this opinion. 2) One is not to novelize things in Torah that have been accepted from previous generations. 3) According to Kabala, the Mitzvah of Techeiles is not applicable in times of exile and will only resurface during the times of Moshiach. [Letter of Rebbe Rashab ibid; Mekubalim in next footnote; See Piskeiy Teshuvos 8 footnote 2]
 Arizal in Peri Eitz Chaim Shaar Hatzitzis 5; Likkutei Torah Shelach 45; Aruch Hashulchan 8/12
 Likkutei Torah Korach “Vayikach Korach” page 104
 Birkeiy Yosef 8/4 in name of the Kabbalist Yaakov Tzemach; See letter of Rebbe Rashab ibid; Arizal ibid