{"product_id":"congruence-clock","title":"Congruence Clock","description":"\u003cp\u003e\u003cstrong\u003eIt's π o' clock!\u003c\/strong\u003e\u003cbr\u003e\u003cstrong\u003e \u003c\/strong\u003e\u003c\/p\u003e\u003cp\u003eThis \u003cstrong\u003eCongruence Clock\u003c\/strong\u003e is no ordinary clock—it's based on \u003cstrong\u003emodular arithmetic\u003c\/strong\u003e, where each number is represented using interesting mathematical expressions that reduce \u003cstrong\u003emod 12 \u003c\/strong\u003eto the expected value. The goal is to get you as well as anyone seeing it interested in some cool math.\u003cbr\u003e \u003c\/p\u003e\u003cp\u003eFor assembly, put the clock in the square on the back, and screw the knut on, from the other side, to hold it fast. I have also included some clock hands in the print profile, on a second plate. There is a hidden reference to a well-known number sequence in them.\u003c\/p\u003e\u003cp\u003e \u003c\/p\u003e\u003cp\u003eA quick explanation of modular arithmetic is that it deals with the remainder after division. Specifically, when we're working “mod 12”, we're dividing by 12 and finding the remainder. For example, 14 mod 12 equals 2 because when you divide 14 by 12, the remainder is 2. Similarly, 27 mod 12 equals 3, since dividing 27 by 12 leaves a remainder of 3. This can also be written as 27 ≡ 3 (mod 12), which is read “27 is congruent to 3, modulo 12”\u003c\/p\u003e\u003ch3\u003e\n\u003cbr\u003eHere are the stories for each number:\u003c\/h3\u003e\u003cp\u003e\u003cstrong\u003e0 → 0\u003c\/strong\u003e\u003c\/p\u003e\u003cp\u003eStarting out simple, with the very important number 0. It plays such a big role in computer science, that it felt right to just leave it as is. Too often 0 is disregarded. Everyone always starts from 1 (but not this time!). Also, there is no number mod 12, that equals 12. In fact 12 mod 12 itself equals 0. \u003cbr\u003e\u003cstrong\u003e1 → e\u003c\/strong\u003e\u003cspan style=\"background-color: rgb(255,255,255); color: rgb(223,0,15)\"\u003e⁰\u003c\/span\u003e,\u003cbr\u003eEulers number “e”, raised to the power of zero is exactly 1. In fact any number to the power of 0 is 1! (Let's not talk about 0^0). This will be your introduction to e (≈ 2.718281828…) - It will appear again later, as it is quite an important number.\u003c\/p\u003e\u003cp\u003e\u003cstrong\u003e2 → !3\u003c\/strong\u003e\u003cbr\u003eThe number of derangements of 3 objects. Place 3 distinct object in front of you in a row, e.g. 3 different colered balls. In how many ways can you move all balls, such that no ball is in the same place it started? The answer is 2.\u003c\/p\u003e\u003cp\u003e\u003cstrong\u003e3 → \u003c\/strong\u003e\u003cspan style=\"background-color: rgb(255,255,255); color: rgb(64,64,64)\"\u003e≈\u003c\/span\u003e\u003cstrong\u003e \u003c\/strong\u003e\u003cspan style=\"background-color: rgb(255,255,255); color: rgb(51,51,51)\"\u003eπ\u003c\/span\u003e\u003c\/p\u003e\u003cp\u003ePi. The lovely mathematical constant related to circles, feels quite at home on a round clock. Pi = 3,141592… which is why the line for pi is slightly below horizontal!\u003c\/p\u003e\u003cp\u003e\u003cstrong\u003e4 → 10\u003c\/strong\u003e\u003cspan style=\"background-color: rgb(255,255,255); color: rgb(43,43,43); font-size: 14px\"\u003eᵏ\u003c\/span\u003e\u003cstrong\u003e (k∈ℕ_{\u0026gt;1})\u003c\/strong\u003e\u003c\/p\u003e\u003cp\u003e10\u003cspan style=\"background-color: rgb(255,255,255); color: rgb(43,43,43); font-size: 14px\"\u003eᵏ\u003c\/span\u003e mod 12 (where k is a whole number, larger than 1) is always equal to 4. So 100, 1000, 10000 and so on mod 12, all are equal to 4. This is the first time the congruence comes into play. The proof of this is left as an exercise to the reader :)\u003c\/p\u003e\u003cp\u003e\u003cstrong\u003e5 → 101\u003c\/strong\u003e\u003c\/p\u003e\u003cp\u003eHave you ever heard of binary numbers? 5 written in binary is “101”. But even if you don't know about binary, 101 mod 12 is still equal to 5. There are only two other positive numbers, which written in binary mod 12 is equal to the number itself! \u003c\/p\u003e\u003cp\u003e\u003cstrong\u003e6 → 2*3*5*…*p\u003c\/strong\u003e\u003cspan style=\"background-color: rgb(255,255,255); color: rgb(223,0,15)\"\u003eₖ\u003c\/span\u003e\u003cstrong\u003e (k∈ℕ_{\u0026gt;1})\u003c\/strong\u003e\u003cbr\u003eThe product of the first k primes mod 12 (where k again is a whole number larger than 1) is always equal to 6. For example 2*3*5*7*11 ≡ 6 (mod 12). This continues for \u003cstrong\u003eany \u003c\/strong\u003enumber of primes you multiply. This is an fun fact i realized while making this clock. See if you can figure out why its true?\u003c\/p\u003e\u003cp\u003e\u003cstrong\u003e7 →\u003c\/strong\u003e\u003c\/p\u003e\u003cp\u003e\u003cstrong\u003e \u003c\/strong\u003e\u003cimg class=\"image_resized\" style=\"width: 5.26%\" src=\"https:\/\/makerworld.bblmw.com\/makerworld\/model\/DSM00000001258678\/design\/2025-03-28_4b3e68f23e624.png\"\u003e\u003cbr\u003eThe determinant of the matrix consisting of 3, i and 2. Taking the determintant of a matrix, is calculated by 3*2 - i*i. Here i is the imaginary unit, where i\u003cspan style=\"background-color: rgb(255,255,255); color: rgb(223,0,15)\"\u003e²\u003c\/span\u003e = -1. So we get 3*2 - (-1) = 6+1 = 7.\u003c\/p\u003e\u003cp\u003e\u003cstrong\u003e8 → |P({\u003c\/strong\u003e\u003cspan style=\"background-color: rgb(255,255,255); color: rgb(0,0,0)\"\u003eℚ\u003c\/span\u003e\u003cstrong\u003e,Ø,ℂ})|\u003c\/strong\u003e\u003cbr\u003eThe size of the powerset of three elements is 8. A powerset is all possibly combinations of the three elements \u003cspan style=\"background-color: rgb(255,255,255); color: rgb(0,0,0)\"\u003eℚ\u003c\/span\u003e Ø and ℂ. In this case it would be Ø, {Ø}, {ℂ}, {\u003cspan style=\"background-color: rgb(255,255,255); color: rgb(0,0,0)\"\u003eℚ\u003c\/span\u003e}, {\u003cspan style=\"background-color: rgb(255,255,255); color: rgb(0,0,0)\"\u003eℚ\u003c\/span\u003e,ℂ}, {\u003cspan style=\"background-color: rgb(255,255,255); color: rgb(0,0,0)\"\u003eℚ\u003c\/span\u003e,Ø}, {Ø,ℂ} and {\u003cspan style=\"background-color: rgb(255,255,255); color: rgb(0,0,0)\"\u003eℚ\u003c\/span\u003e,Ø, ℂ}. Ø is the empty set, ℂ is the set of complex numbers, while \u003cspan style=\"background-color: rgb(255,255,255); color: rgb(0,0,0)\"\u003eℚ\u003c\/span\u003e is the set of rational numbers. \u003c\/p\u003e\u003cp\u003e \u003c\/p\u003e\u003cp\u003e\u003cstrong\u003e9 →\u003c\/strong\u003e\u003c\/p\u003e\u003cp\u003e\u003cimg class=\"image_resized\" style=\"width: 6.63%\" src=\"https:\/\/makerworld.bblmw.com\/makerworld\/model\/DSM00000001258678\/design\/2025-03-28_1a929738dff6c.png\"\u003e\u003c\/p\u003e\u003cp\u003eThree to the power of (3 tetrated, plus 1). Writing the exponent before the number is notation for tetration. It is another way of writing two of knuths arrows, for those who know them. This is also the same of writing 3^3^3. So what it really says is 3^(3^3^3 +1). Now 3^(3^3^3 +1) ≡ 9 (mod 12). The proof of this if left to the reader (Hint: Show that 3^(2*k) ≡ 9 (mod 12), where k is a positive integer)\u003c\/p\u003e\u003cp\u003e\u003cstrong\u003e10 → A\u003c\/strong\u003e\u003cspan style=\"background-color: rgb(255,255,255); color: rgb(223,0,15)\"\u003e₁₆\u003c\/span\u003e\u003c\/p\u003e\u003cp\u003eBack to computer science. In hexadecimal notation A\u003cspan style=\"background-color: rgb(255,255,255); color: rgb(223,0,15)\"\u003e₁₆ (or \u003c\/span\u003e0xA) represents 10. The system goes 1, 2, 3, … 9, A, B, … F, where F is 15. You might have seen this in hex-codes for colors. Eg. 0xFFFFFF is white.\u003c\/p\u003e\u003cp\u003e\u003cstrong\u003e11 → e^(\u003c\/strong\u003e\u003cspan style=\"background-color: rgb(255,255,255); color: rgb(51,51,51)\"\u003eπ\u003c\/span\u003e\u003cstrong\u003e*i)\u003c\/strong\u003e\u003c\/p\u003e\u003cp\u003eEulers identity. Would this really be a math clock without it? Eulers identity is quite famous in math, combining all our favorites, e, \u003cspan style=\"background-color: rgb(255,255,255); color: rgb(51,51,51)\"\u003eπ\u003c\/span\u003e and i. Euler found that e^(\u003cspan style=\"background-color: rgb(255,255,255); color: rgb(51,51,51)\"\u003eπ\u003c\/span\u003e*i) = -1 (sometimes written e^pi*i - 1 = 0). Combine this with  -1 ≡ 11 (mod 12) and it works as 11 on the clock.\u003c\/p\u003e\u003cp\u003e \u003c\/p\u003e\u003cp\u003e \u003c\/p\u003e\u003ch3\u003eThat's it!\u003c\/h3\u003e\u003cp\u003eIf you are really curious on some of the proofs, and can't figure it out, write a comment (or see if someone has already asked) and i'll give you mine!\u003cbr\u003e\u003cbr\u003eIf you spot any errors, please do tell and I'll fix them!\u003c\/p\u003e\u003cp\u003eDesign by Buildasaurus on MakerWorld (license: BY-SA).\u003c\/p\u003e","brand":"Mymadmanlab","offers":[{"title":"Default Title","offer_id":67447038116144,"sku":null,"price":6.99,"currency_code":"USD","in_stock":true}],"thumbnail_url":"\/\/cdn.shopify.com\/s\/files\/1\/0987\/0809\/5280\/files\/0_323f4e6a-1ec1-48d8-9448-0a051bbffdbd.png?v=1785813304","url":"https:\/\/mymadmanlab.com\/products\/congruence-clock","provider":"Mymadmanlab.com","version":"1.0","type":"link"}