A fractal is a mathematical set that has a fractal dimension that usually exceeds its topological dimension and may fall between the integers. Fractals are typically self-similar patterns, where self-similar means they are "the same from near as from far". Fractals may be exactly the same at every scale or they may be nearly the same at different scales. The definition of fractal goes beyond self-similarity per se to exclude trivial self-similarity and include the idea of a detailed pattern repeating itself.
The Mandelbrot set is a mathematical set of points whose boundary is a distinctive and easily recognizable two-dimensional fractal shape. The set is closely related to Circular Julia sets (which include similarly complex shapes), and is named after the mathematician Benoit Mandelbrot, who studied and popularized it.
Images of the Mandelbrot set are made by taking numbers on the complex plane, calculating whether it tends to infinity when the formula is iterated on the number, then using the number as X and Y coordinates in the picture and coloring the pixel depending on whether it tends to infinity or not.
Highlighting ever-changing, spiraling colors and shapes on the sphere get students excited to learn more about calculus. They are also great for parties under the sphere!
C1 Patterns. Children recognize that patterns in the natural and human designed world can be observed, used to describe phenomena, and used as evidence
C1 Patterns. Students identify similarities and differences in order to sort and classify natural objects and designed products. They identify patterns related to time, including simple rates of change and cycles, and to use these patterns to make predictions.
C3 Scale Proportion and Quantity. Students recognize natural objects and observable phenomena exist from the very small to the immensely large. They use standard units to measure and describe physical quantities such as weight, time, temperature, and volume.
C1 Patterns. Students recognize that macroscopic patterns are related to the nature of microscopic and atomic-level structure. They identify patterns in rates of change and other numerical relationships that provide information about natural and human designed systems. They use patterns to identify cause and effect relationships, and use graphs and charts to identify patterns in data.
C4 Systems and System Models. Students can understand that systems may interact with other systems; they may have sub-systems and be a part of larger complex systems. They can use models to represent systems and their interactions—such as inputs, processes and outputs—and energy, matter, and information flows within systems. They can also learn that models are limited in that they only represent certain aspects of the system under study.
C4 Systems and System Models. Students can investigate or analyze a system by defining its boundaries and initial conditions, as well as its inputs and outputs. They can use models (e.g., physical, mathematical, computer models) to simulate the flow of energy, matter, and interactions within and between systems at different scales. They can also use models and simulations to predict the behavior of a system, and recognize that these predictions have limited precision and reliability due to the assumptions and approximations inherent in the models. They can also design systems to do specific tasks.
LS1.D Information Processing. Animals sense and communicate information and respond to inputs with behaviors that help them grow and survive.
PS4.C Information Technologies and Instrumentation. People use devices to send and receive information.
LS1.D Information Processing. Different sense receptors are specialized for particular kinds of information; Animals use their perceptions and memories to guide their actions.
PS4.C Information Technologies and Instrumentation. Patterns can encode, send, receive and decode information.
LS1.D Information Processing. Each sense receptor responds to different inputs, transmitting them as signals that travel along nerve cells to the brain; The signals are then processed in the brain, resulting in immediate behavior or memories.
PS4.A Wave Properties. A simple wave model has a repeating pattern with a specific wavelength, frequency, and amplitude, and mechanical waves need a medium through which they are transmitted. This model can explain many phenomena including sound and light. Waves can transmit energy
PS4.C Information Technologies and Instrumentation. Waves can be used to transmit digital information. Digitized information is comprised of a pattern of 1s and 0s.
PS4.A Wave Properties. The wavelength and frequency of a wave are related to one another by the speed of the wave, which depends on the type of wave and the medium through which it is passing. Waves can be used to transmit information and energy.
PS4.B Electromagnetic Radiation. Both an electromagnetic wave model and a photon model explain features of electromagnetic radiation broadly and describe common applications of electromagnetic radiation.
PS4.C Information Technologies and Instrumentation. Large amounts of information can be stored and shipped around as a result of being digitized.