Several studies, such as “Seeing and feeling volumes: The influence of shape on volume perception”, have shown that people have a tendency to overestimate the volume of common objects, such as glasses and containers, that are tall and thin and underestimate those that are short and wide; this phenomenon is called “elongation bias”. 

Sue Palmberg, an instructor at Edwin O. Smith High School, created and shared with us a lab activity for students to design a glass in Maple and use volumes of revolution to determine the amount of liquid it can hold. This lab was then turned into this Maple Learn document: Piecewise Volumes of Revolution Activity.

Use this document to create your own glass or goblet shape and determine its volume. Simply create a piecewise function that will define the outside shape of your glass between your chosen bounds and another piecewise function to define the hollowed-out part of your creation. The document will graph the volumes of revolution that represent your glass and calculate the relevant volume integral for you.

Here is my own goblet-shaped creation: 

I used this piecewise function to define it:

After creating the outline of my goblet, I constructed a function for the hollow part of the goblet – the part that can actually hold liquid.

Using Context Panel operations and the volume integral provided by the document, I know that the volume of the hollow part of my goblet is approximately 63.5, so my goblet would hold around 63.5 units³ of liquid when full.

Create your own goblets of varying shapes and see if their volumes surprise you; elongation bias can be tricky! For some extra help, check out the Piecewise Functions and Plots and Solids of Revolution - Volume Derivation documents!

The recent Maple 2023 release comes with a multitude of new features, including a new Canvas Scripting Gallery full of templates for creating interactive Maple Learn documents.

The Maple Learn Scripting Gallery can be accessed through Maple, by searching “BuildInteractiveContent Maple2023” in the search bar at the top of the application and clicking on the only result that appears. This will bring you to the help page titled “Build and Share Interactive Content”, which can also be found by searching “scripting gallery” in the search bar of a Maple help page window. The link to the Maple Learn Scripting Gallery is found under the “Canvas Scripting” section on this help page and clicking on it will open a Maple workbook full of examples and templates for you to explore.

The interactive content in the Scripting Gallery is organized into five main categories – Graphing, Visualization, Quiz, Add-ons and Options, and Applications Optimized for Maple Learn – each with its own sub-categories, templates, and examples.

One of the example scripts that I find particularly interesting is the “Normal Distribution” script, under the Visualizations category.

All of the code for each of the examples and templates in the gallery is provided, so we can see exactly how the Normal Distribution script creates a Maple Learn canvas. It displays a list of grades, a plot for the grade distribution to later appear on, math groups for the data’s mean and variance, and finally a “Calculate” button that runs a function called UpdateStats.

The initial grades loaded into the document result in the below plot, created using Maple’s DensityPlot and Histogram functions, from the Statistics package. 

The UpdateStats function takes the data provided in the list of grades and uses a helper function, getDist, to generate the new plot to display the data, the distribution, the mean, and the variance. Then, the function uses a Script object to update the Maple Learn canvas with the new plot and information.

The rest of the code is contained in the getDist function, which uses a variety of functions from Maple’s Statistics package. The Normal Distribution script takes advantage of Maple’s ability to easily calculate mean and variance for data sets, and to use that information to create different types of random variable distributions.

Using the “Interactive Visualization” template, provided in the gallery, many more interactive documents can be created, like this Polyhedra Visualization and this Damped Harmonic Oscillator – both from the Scripted Gallery or like my own Linear Regression: Method of Least Squares document.

Another new feature of Maple 2023 is the Quiz Builder, also featured in the Scripting Gallery. Quizzes created using Quiz Builder can be displayed in Maple or launched as Maple Learn quizzes, and the process for creating such a quiz is short.

The QuizBuilder template also provides access to many structured examples, available from a dropdown list:

As an example, check out this Maple Learn quiz on Expected Value: Continuous Practice. Here is what the quiz looks like when generated in Maple:

This quiz, in particular, is “Fill-in the blank” style, but Maple users can also choose “Multiple Choice”, “True/False”, “Multiple Select”, or “Multi-Line Feedback”. It also makes use of all of the featured code regions from the template, providing functionality for checking inputted answers, generating more questions, showing comprehensive solutions, and providing a hint at the press of a button.

Check out the Maple Learn Scripting Gallery for yourself and see what kinds of interactive content you can make for Maple and Maple Learn!

When introduced to geometry, one of the first things we learn is the definition of the word “polygon”. A polygon is a closed 2-dimensional shape with at least 3 straight sides and angles. A regular polygon is a polygon with congruent sides and equal angles. A regular polygon with n sides has Schläfli symbol {n}. I’m interested in mathematical history, so when I learned that the idea of higher-dimensional spaces was invented in the middle of the nineteenth century I decided to research more about Ludwig Schläfli and the notation he came up with to describe his ideas.

In general, the Schläfli symbol is a notation of the form {p, q, r, ...} for regular polytopes. Polytopes are geometric objects with flat sides. This week, I will be focusing on 3-dimensional polytopes, also called polyhedra.

Similar to regular polygons, regular polyhedra are 3-dimensional shapes whose faces are all the same regular polygon. A regular polyhedron’s Schläfli symbol is of the form {p, q}, where p is the number of edges each face has and q is the number of faces that meet at each of the polyhedron’s vertices.

Below are two regular polyhedra: a cube (also known as a hexahedron) and a great stellated dodecahedron. The cube is one of five Platonic solids, and the great stellated dodecahedron is one of four Kepler-Poinsot polyhedra – all of these can be represented by Schläfli symbols. The cube has Schläfli symbol {4, 3}, since squares have 4 equal sides, and each vertex of a cube is created by the vertices of 3 squares meeting.

Can you figure out the Schläfli symbol for the great stellated dodecahedron?

The great stellated dodecahedron has the Schläfli symbol {5/2, 3}. This is because great stellated dodecahedrons are regular star polygons. As a result, the first number in their Schläfli symbol is an irreducible fraction whose numerator represents a number of sides and whose denominator corresponds to a turning number. The particular fraction 5/2 corresponds to a pentagram – a regular star polygon with 5 points – and great stellated dodecahedrons are composed of 12 of these pentagrams, where 3 pentagrams meet at each vertex of the shape.

One notable example of a regular polytope in pop culture is the tesseract, which has the Schläfli symbol {4, 3, 3}. This is an extension of the cube’s Schläfli symbol, {4, 3}, and the last number indicates that there are three cubes folded together around every edge. Below are two representations of a tesseract: one that uses a Schlegel diagram (left) and one from the 2012 movie Avengers (right).

Try out our Regular Polyhedra Visualization Using Schlafli Symbol Notation! In this document, you can test out your own Schläfli symbols for regular polyhedra. If they are valid Schläfli symbols, you’ll be provided with a 3-D visualization of the shape. If they are invalid, you can check out the logic for finding the specifications for regular polyhedra and this document, which provides all the 3-D regular polyhedra for you to try out.