While the real answer is the one posted by Shmuel, I’d like to post an answer in the line of your final question, “Are there any other terms and expressions commonly shared?”
φ Fi Fo Fum
First, I’d like to address your example of φ, the Golden Ratio. The reason those two numbers have anything to do with φ is because they are in the Fibonacci sequence (1, 1, 2, 3, 5, 8, 13, etc., where each term is the sum of the previous two terms); it’s well-known (certainly amongst mathematicians, and to a great extent among non-mathematicians as well) that if you take any number in this list and divide it by the previous one, it will give an approximation of φ=1.618..., with the approximation getting more and more accurate as you get closer to infinity (in mathspeak, the series converges to φ). You can verify this with a calculator, but the twelfth Fibonacci number is 144, and the thirteenth is 233; thus, following the pattern stated above, 233/144 is approximately φ. (What this has anything to do with Gan Eden I will leave as an exercise to the reader.)
π in the Sky
Now, for something a bit different, but in the same vein as the example above. The following is often ascribed to the Vilna Gaon, but it seems that it’s actually from an article by one Rabbi Munk in the journal Sinai, from 1962.
The Talmud in Eruvin 14a derives that, for halachic purposes,1 we can round off the ratio of the circumference of a circle to its diameter as π=3, from the following verse, in I Kings 7:23:
וַיַּ֥עַשׂ אֶת־הַיָּ֖ם מוּצָ֑ק עֶ֣שֶׂר בָּ֠אַמָּה מִשְּׂפָת֨וֹ עַד־שְׂפָת֜וֹ עָגֹ֣ל ׀ סָבִ֗יב וְחָמֵ֤שׁ בָּֽאַמָּה֙ קוֹמָת֔וֹ וקוה [וְקָו֙] שְׁלֹשִׁ֣ים בָּֽאַמָּ֔ה יָסֹ֥ב אֹת֖וֹ סָבִֽיב׃
And [Solomon] made the Sea2 of cast metal. [It was] ten cubits from one lip to the other lip, circular all around; its height was five cubits, and a line of thirty cubits could surround it.
I’ve translated this verse as literally as I can, with the first set of brackets replacing “he” with “Solomon” for context. The important part here is that it had a diameter of ten cubits and a circumference of thirty, yielding a ratio of π=3.
The interesting part comes in when you notice that, in this verse - but not its parallel in II Chronicles 4:2 - the word וקו, “and a line,” is pronounced as if it were spelled that way, but it’s written as וקוה. In fact, this particular way it’s written is nearly unique to this verse among the sixteen places where the word appears (there are two others).
The vav at the beginning of the word is just a conjunction, so we drop that, leaving the base words קו and קוה. The latter has a numerical value of 111, and the former has a numerical value of 106. Multiplying this ratio by three gives a pretty decent estimate of π=3.14150943; subtracting this from actual π yields an error of <0.000084!
What’s more impressive is that, among the denominators under 30,000, this is the second-best approximation there is (but it’s a really close second). This corresponds to the fact that, in the continued fraction for π, 3+(1/(7+1/(15+1/(1+..., the approximation 3*111/106 is equivalent to the third layer of the infinitely expanding fraction. (I apologize for the messiness; these are the days I wish MY supported MathJax.)
For further reading on the topic, I recently came across this article, which seems to be quoting this one, which details some of the points I mentioned above, as well as some others. The former, on the other pages on the site, has some other fascinating estimations of π from other places in the Bible.
1Technically, this interpretation of the Gemara follows the Tosfos HaRosh there; the Aruch HaShulchan seems to learn that Gemara differently, in which we can only interpret this approximation as being relevant to Eiruvin. I follow Tosfos HaRosh here simply because it’s easier to explain the Gemara, IMO, according to his reading.
2In context, the “Sea of Solomon” refers to a basin he built in the Temple used by the priests for washing their hands and feet before serving. As derived by the Talmud quoted above, the Sea was cylindrical at the top and rectangular toward the bottom.