At the conclusion of Parshas Shlalch, the Mitzvah of Tzitzis is discussed within the well known passage of our daily קריעת שמע . The Torah explicitly states the connection of seeing the "tzitzis" and remembering all of the Mitzvos. "והיה לכם לציצת וּראיתם אתו וּזכרתם את כּל מצות ה'" You shall see them (Tzitzis) and remember all of the Mitzvos. The question is how? In what way?
There is a debate amongst the ראשונים as to what about this Mitzvah brings about the consciousness or remembrance of all other mitzvos. רש"י is of the opinion that through a combination of the numerical value of the word ציצית, plus the five knots and eight strings - the sum will equal 613, the number of all the mitzvos. The Ramban, however, takes issue with this methodology and offers an alternative approach to the connection ofציצית and it's remembrance of all the מצות. The Ramban states that the remembrance comes through the"חוט התכלת", the bluish-turquoise strings that are placed together with the other white strings. (This approach happens to fit appropriately in both context and order of the פסוקים. The פסוק of וזכרתם את כל מצות ה'immediately following the פסוק of פתיל תכלת.) The Ramban proceeds to quote, in support of his approach, the גמרא (Menachos 43b) which describes how the color of תכלת is similar to the sea, and the sea is similar to the sky, and once being cognizant of the sky, one will give thought to the כסא הכבוד on High, which will, in-turn, remind him of all the Mitzvos.
One may ask himself: why does the Talmud list these specific examples? The sea, the sky. Wouldn't the fact that blue reminds a person of the sky be sufficient to get him to be aware of the כסא הכבוד ? Furthermore, why blue? Why not the color white which is the same color as the clouds in the sky? Finally, why is it called תכלת and not כחול?
Let us analyze these two examples for a moment. If one were to ask, what color the sky or the sea is, the answer would obviously be "blue" (on a clear day). However, the fact of the matter is, both of them are in-fact not inherently blue, but rather clear and completely transparent with no trace of any color. Even if one were a reflection of the other, how then did the other get blue? So where does this color of blue come from?
Interestingly, from a scientific perspective, one would learn that light, which is made up of all our colors, travels in a straight, direct line except in three distinct scenarios: a prism (where it's bent), a mirror (where it's reflected), and when it comes in contact with this world's atmosphere (where it's scattered). In the latter scenario, light's contact to this world, the color blue from the color spectrum is manifest upon the sun-light's contact with the atmosphere (blue light is scattered the quickest because of its rapid wave sequence). That is why we see the sky as blue.
The Talmud states (פסחים ב:) that this world is compared to the night, darkness. The פסוק in משלי also says ״נר מצוה ותורה אור״ the Mitzvos are our candle and the Torah is our light. Without a light shining the path in front of us we would be blindly stumbling over the many different obstacles positioned right in front of our feet. Our purpose here is to bring in that light, that אור, through the Torah and Mitzvos, to spiritually shine and light up a world that is compared to darkness.
The Mitzvah of Tzitzis serves as a reminder for these other Mitzvos. How? Precisely from theתכלת string, as the רמב"ן says. Just like when light shines through into this world, earth's atmosphere, what's manifest in the sky is the color blue, so too, we are commanded to put the blue strings - which represent this idea of bringing light into the world, together with the white strings, on our בגד to serve as a reminder for a very fundamental purpose in this world, namely, to shine forth Light, אור , thru the מצות, in a world that's dark.
How apropos that the ספר החינוך writes (mitzvah 326) that the white fringes of Tzitzis represent the physical whereas the blue represents the spiritual, as it echoes this very idea of bringing spiritual light, from Mitzvos, into a physical world.
This may be what חז״ל were alluding to with these examples of sea and sky. Neither are inherently blue, but rather mirror that color as a result of the light shining into this world.
Now it is very understandable why the חוט התכלת of Tzitzis serves as a reminder for all other מצות, according to the רמב"ן. After the רמב"ן gives this approach, he then notes that this word תכלת is also the word תכלית, purpose. That string serves as the reminder for our very fundamental purpose in this world.