Recursive Roots Slides RECURSIVE ROOTS
The Algebra of Iteration
The Infinite Feedback Loop
Imagine a number. Square it. Add 1. Take that answer and do it again.
\( f(x) = x^2 + 1 \)
"What happens if we repeat this process 10 times? 100 times? Forever?"
What is Iteration?
Recursive Definition
A process where the output of one step becomes the input for the next.
Mathematical Notation
\( z_{n+1} = z_n^2 + c \)
Where \( z_0 \) is our starting value.
// Real Number Example
Let \( z_0 = 0 \), \( c = 1 \)
\( z_1 = 0^2 + 1 = 1 \)
The Complex Upgrade
Now, we replace real numbers with Complex Numbers.
Remember: \( i^2 = -1 \)
Target Calculation: \( z_{n+1} = z_n^2 + i \)
Start \( z_0 = 0 \)
Iter 1 \( z_1 = 0^2 + i = i \)
Iter 2 \( z_2 = i^2 + i = -1 + i \)
Iter 3 \( z_3 = (-1 + i)^2 + i \)
Wait... how do we square \( (-1 + i) \)?
Algebraic Combat: Squaring Binomials
The Formula
\( (a + bi)^2 \)
\( a^2 + 2abi + (bi)^2 \)
\( (a^2 - b^2) + 2abi \)
Let's solve Iteration 3:
\( z_3 = (-1 + i)^2 + i \)
Expand the square:
\( = (1 - 2i + i^2) + i \)
\( = (1 - 2i - 1) + i \)
\( = -2i + i \)
\( = -i \)
Your Turn
Calculate the first 3 iterations of:
\( z_{n+1} = z_n^2 + (1 + i) \)
Initial Value: \( z_0 = 0 \)
Grab your lab reports and start the sequence!
Iteration Lab Worksheet ITERATION LAB REPORT
SUBJECT: COMPLEX DYNAMICS // SERIES 01
Agent:
Date:
Mission Briefing
Iteration is the heart of fractal geometry. To map the complex plane, we must master the recursive process. Today, we focus on the rule: \( z_{n+1} = z_n^2 + c \), where \( z_n \) is our current point and \( c \) is a complex constant.
1 REAL NUMBER WARMUP
Calculate the first four terms for \( z_{n+1} = z_n^2 + (-2) \) starting at \( z_0 = 0 \).
\( z_1 \)
\( z_0^2 - 2 = \)
\( z_2 \)
\( z_1^2 - 2 = \)
\( z_3 \)
\( z_2^2 - 2 = \)
\( z_4 \)
\( z_3^2 - 2 = \)
2 COMPLEX PLANE DIVE
Perform 3 iterations for the following complex constants. Show all work, especially your binomial expansions! Assume \( z_0 = 0 \) for all tasks.
Constant A: \( c = 2i \) Show Your Work Below
Iteration 1 (\( z_1 \))
Iteration 2 (\( z_2 \))
Iteration 3 (\( z_3 \))
Constant B: \( c = 1 - i \) Show Your Work Below
Iteration 1 (\( z_1 \))
Iteration 2 (\( z_2 \))
Post-Lab Analysis
Look at your results for Constant A and Constant B. In 1-2 sentences, what is happening to the size of the numbers as you iterate? Are they getting smaller, larger, or staying the same?
Iteration Teacher Guide FACILITATION GUIDE
Lesson 1: Recursive Roots
Instructional Intent
This lesson bridges the gap between static algebra and iterative dynamics. Students often view variables as fixed values; here, they see them as evolving states. The primary technical goal is fluency with complex arithmetic, specifically squaring binomials involving \( i \).
Essential Question
How do repeated simple operations create algebraic change?
Key Vocabulary
Iteration: The act of repeating a process.
Recursion: Defining a process in terms of itself.
Seed Value: The initial input (\( z_0 \)).
Constant: The fixed value (\( c \)).
Lesson Flow & Pacing
10 min
The Feedback Loop Hook
Use Slide 2 to introduce the concept. Ask: "If we square 0.5 and add 1 repeatedly, where does it go? What if we square 2 and add 1?" This primes them for the concept of growth vs. stability .
15 min
Technical Refresher
Slides 4-5 focus on the squaring of \( (a + bi) \). This is where students will trip up. Emphasize that \( i^2 = -1 \) and it always changes the sign of the constant term.
Pro-tip: Watch for the common error \( (a+bi)^2 = a^2 - b^2 \). Remind them the middle term \( 2abi \) is essential!
20 min
The Lab Report
Distribute the Iteration Lab Worksheet . Circulate and check for arithmetic precision. Encourage students to check each other's work after each iteration step.
Internal Answer Key
Constant A: \( c = 2i \)
\( z_1 = 0^2 + 2i = \mathbf{2i} \)
\( z_2 = (2i)^2 + 2i = -4 + 2i \)
\( z_3 = (-4 + 2i)^2 + 2i = (16 - 16i - 4) + 2i = \mathbf{12 - 14i} \)
Constant B: \( c = 1 - i \)
\( z_1 = 0^2 + (1-i) = \mathbf{1 - i} \)
\( z_2 = (1-i)^2 + (1-i) = (-2i) + 1 - i = \mathbf{1 - 3i} \)
Check: \( (1-i)^2 = 1 - 2i - 1 = -2i \)
Debrief Questions
"Did anyone find a number that stayed the same or returned to 0?" (Foreshadowing cycles)
"Which constant made the numbers grow faster? Why?" (Connecting to absolute value)
Orbit Mapper Slides ORBIT MAPPER
Tracking the Flight of Numbers
The Mathematical Orbit
An orbit is the sequence of points generated by iterating a function.
Iteration Sequence
\( z_0, z_1, z_2, z_3, ... \)
Visual Orbit
The path formed by connecting these points in the complex plane.
Mapping the Plane
Coordinates
\( a + bi \rightarrow (a, b) \)
Real axis (\( x \))
Imaginary axis (\( y \))
Complex Plane View
Case Study: The Cycler
\( c = -1 \)
\( z_0 = 0 \)
\( z_1 = 0^2 - 1 = -1 \)
\( z_2 = (-1)^2 - 1 = 0 \)
\( z_3 = 0^2 - 1 = -1 \)
"Trapped in a feedback loop"
0
-1
Case Study: The Explosion
\( c = 1 \)
\( z_0, z_1, z_2, z_3... \)
\( 0, 1, 2, 5, 26, 677... \)
This orbit escapes to infinity. It grows faster and faster with every step.
Mission: Pathfinder
You will be assigned unique complex constants. Your job:
1 Calculate the first 5 terms.
2 Plot them on the grid.
3 Connect them with arrows.
Observation Goal
"Does your number fly away, get trapped, or dance around the origin?"
Orbit Mapping Sheet Mission: Pathfinder
ORBIT MAPPING SHEET
Assigned Constant:
\( c = \) _______
1 CALCULATE THE FLIGHT PATH
Rule: \( z_{n+1} = z_n^2 + c \) // Start at \( z_0 = 0 \)
Step Work Area Result (\( z_n \)) \( z_0 \) Initial Condition 0 \( z_1 \) \( z_2 \) \( z_3 \) \( z_4 \)
2 CHART THE ORBIT
Imaginary (\( i \))
Real
START (0,0)
Connect your points with arrows to show sequence
OBSERVATION LOG
Distance from Origin:
Are the points getting further away from (0,0) as you iterate, or do they stay close?
Classification Prediction:
If you were to repeat this 100 times, would your number reach infinity or stay bounded?
COLLABORATION: Compare your map with a neighbor. Do your orbits behave similarly?
Orbit Pathfinder Solutions Key ORBIT MAP SOLUTIONS
Teacher Answer Key // Lesson 2
Grading Note
The "Pathfinder" activity assigns different constants to students. Below are the calculations for 4 common constants representing the 4 main behaviors (Escaping, Cyclic, Fixed, and Chaotic).
Type A: The Escaper
\( c = 1 + i \)
\( z_0 = 0 \)
\( z_1 = 1 + i \)
\( z_2 = (1+i)^2 + (1+i) = 2i + 1 + i = \mathbf{1 + 3i} \)
\( z_3 = (1+3i)^2 + (1+i) = (1 - 9 + 6i) + 1 + i = \mathbf{-7 + 7i} \)
\( z_4 = (-7+7i)^2 + (1+i) = (49 - 49 - 98i) + 1 + i = \mathbf{1 - 97i} \)
Visual behavior
The orbit spirals outward rapidly. By \( z_4 \), the magnitude exceeds the standard plotting grid (Distance \(\approx 97\)).
Type B: The Cycler
\( c = -1 \)
\( z_0 = 0 \)
\( z_1 = -1 \)
\( z_2 = 0 \)
\( z_3 = -1 \)
\( z_4 = 0 \)
Visual behavior
The orbit oscillates between 0 and -1 on the real axis. This is a 2-cycle orbit. It is perfectly bounded.
Type C: The Prisoner
\( c = i \)
\( z_0 = 0 \)
\( z_1 = i \)
\( z_2 = i^2 + i = -1 + i \)
\( z_3 = (-1+i)^2 + i = -2i + i = -i \)
\( z_4 = (-i)^2 + i = -1 + i \)
Visual behavior
After \( z_1 \), the orbit enters a 2-cycle between \(-1+i\) and \(-i\). It stays near the origin forever.
Common Misconceptions to Watch For:
The Square Trap: Students often calculate \( (a+bi)^2 \) as \( a^2 + b^2i^2 \). Remind them that it is a binomial expansion: \( a^2 + 2abi - b^2 \).
Axis Confusion: Students may plot \( 1+3i \) as \( (3, 1) \) instead of \( (1, 3) \). Always verify the horizontal axis is "Real".
Prisoners of Infinity Slides PRISONERS OF INFINITY
Defining the Mandelbrot Set
Orbit Destinations
Escaping
The orbit grows larger and larger. Eventually, its distance from the origin passes a point of no return.
Destiny: Infinity (\( \infty \))
Prisoner
The orbit stays close to the origin. It might cycle, sit still, or bounce around, but it never leaves.
Destiny: Bounded Set
The Threshold of Escape
Mathematics tells us that if an orbit ever passes a distance of 2 from the origin, it will always escape to infinity.
Distance Check (Modulus):
\( |z| = \sqrt{a^2 + b^2} \)
If \( |z| > 2 \), it's over.
Radius = 2
Infinity Zone
The Mandelbrot Set
"The Mandelbrot set consists of all complex constants c for which the orbit starting at \( z_0 = 0 \) is bounded."
In other words: The list of constants whose orbits never pass the distance of 2.
Concept: Escape Velocity
Not all "Escapers" are the same.
Some numbers fly away on the first step.
Others take 50 steps of spinning before they finally cross the boundary.
We color points based on how long they take to escape. This creates the famous fractal colors!
1 Iteration
5 Iterations
50+ Iterations
Mission: Escape Velocity
Classify your constants:
Escaper
\( |z_n| > 2 \) at some point
M-Set Member
\( |z_n| \leq 2 \) forever
