Contemporary Mathematics

# Projects

### Projects

Encryption began at least as far back as the Roman Empire. During the reign of Caesar, a particular cypher was used, fittingly named the Caesar Cypher. This encryption process granted the Romans a great tactical advantage. Even if a message was intercepted, it would not make sense to the person intercepting the message.

Find four instances when encryption was used and cracked over the course of history.

## The Golden Ratio in Art and Architecture

The golden ratio has been used in art and architecture as far back as ancient Greece (possibly further). It also appears in South America (Incan architecture). Find five instances of the use of the golden ratio in art or architecture and describe its use in each of those instances.

Budgeting either is, or will shortly be, an important aspect of your life. Managing money well reduces stress in your life, and provides space for planning for future expenses, such as vacations or home improvements.

Imagine your life 10 years from now. Estimate your monthly income. Identify expenses you will encounter monthly (mortgage or rent, car payment, insurance, entertainment, etc.). Decide on an amount you plan to save monthly (this is treated as an expense). Create a spreadsheet with those values. Record your monthly net income (your income minus your expenses). Determine how much money you will have saved over the course of 5 years (ignore interest). Write a reflection on your anticipated financial health.

## Estimating Pi

The value of pi is the ratio of the circumference of a circle to the diameter of the circle. It is also equal to the ratio of the area of the circle to the square of the radius of the square.

Research three ways to physically estimate pi.

Estimate pi using all three processes you found.

Present your process and solutions in class.

## Design Your Own Shift Cypher

A cypher is a message written in such a way as to mask its contents. Changing a message into its cypher form is called encryption. Decryption or deciphering is the process of changing a cyphertext message back into the original (legible) message. One process of encryption is to scramble the letters, symbols, and punctuation of a message according to a mathematical rule. One rule that could be used for such a cypher is addition in a chosen modulus. In this project, you will create such a cypher, encrypt a message, and then decrypt the message.

Step 1: Choose the letters, symbols, and punctuation marks you want to allow in your messages. This should include at least the uppercase letters and a space character. This is your character set.

Step 2: Count the number of characters you will use. Label this number $nn$.

Step 3: Pair each character of your character set an integer from 0 to $(n−1)(n−1)$. Do not assign more than one character to an integer.

Step 4: Choose an integer between 1 and $(n−1)(n−1)$. This will be the number used to create the cypher. Label this number $ss$.

Step 5: Write a message using your character set.

Step 6: Replace every character in your message by the integer with which it was paired in Step 3.

Step 7: For every number, $xx$, from Step 7, perform the addition $x+sx+s$ (mod $nn$).

Step 8: Replace every number found in Step 7 with the character with which it was paired in Step 3. This is your cyphertext.

To decrypt your cyphertext, reverse the steps above.

Step 1: Replace the cyphertext characters with the paired values.

Step 2: For each value $xx$, perform the subtraction $x–sx–s$ (mod $nn$).

Step 3: Replace the numbers from Step 2 with their paired characters from the character set.

The message is then deciphered.

## Design Your Own Cypher Using Multiplication

A cypher is a message written in such a way as to mask its contents. Changing a message into its cypher form is called encryption. Decryption or deciphering is the process of changing a cyphertext message back into the original (legible) message. One process of encryption is to scramble the letters, symbols, and punctuation of a message according to a mathematical rule. One rule that could be used for such a cypher is multiplication in a chosen modulus. In this project, you will create such a cypher, encrypt a message, then decrypt the message.

Step 1: Choose the letters, symbols, and punctuation marks you want to allow in your messages. This should include at least the uppercase letters and a space character. This is your character set.

Step 2: Count the number of characters you will use. Label this number $nn$.

Step 3: Pair each character of your character set an integer from 0 to $(n−1)(n−1)$. Do not assign more than one character to an integer.

Step 4: Choose an integer, labeled $ss$, between 1 and $(n−1)(n−1)$ so that $GCF(n,s)=1GCF(n,s)=1$. This will be the number used to create the cypher.

Step 5: Write a message using your character set.

Step 6: Replace every character in your message by the integer with which it was paired in Step 3.

Step 7: For every number, $xx$, from Step 6, perform the multiplication $x−sx−s$ (mod $nn$).

Step 8: Replace every number found in Step 7 with the character with which it was paired in Step 3. This is your cyphertext.

Before beginning to decrypt in this cypher, you need to know the multiplicative inverse of the value you chose as s.

Step 1: The multiplicative inverse of $ss$ is the number that, when multiplied by $ss$ in your modulus, equals 1. To find this, you will have to multiply $ss$ and every number between 2 and ($n–1n–1$) until the product is 1 (mod $nn$). Once this number is found, the message can be decrypted. Call this number $rr$.

Step 2: To decrypt your cyphertext, replace the cyphertext characters with the paired values.

Step 3: For each of the value, $xx$, perform the multiplication $x×r(modn)x×r(modn)$.

Step 4: Replace the numbers from Step 3 with their paired characters from the character set.

The message is then deciphered.

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