Home > General Resources > Advanced Sketch Gallery

## Site Map

The Advanced Sketch Gallery is a collection of sketches featuring advanced mathematical concepts, custom tools, and striking visualizations, created by Sketchpad users around the world. They are arranged loosely by topic:

The sketches are all available for dowload for your personal or classroom use, but are copyrighted by their authors; please contact the author of a sketch if you are interested in using it in other contexts.

## Toolkits

Toolkit documents contain collections of related custom tools you may find useful in your own explorations.

### Knot Toolkit

This toolkit showcases several example knots, provides some exercises for the reader, and supplies both a knot grid and the tools for constructing your own knots.

Doris Schattschneider, 2002

### Graphing Toolkit

This set of tools and accompanying Instructional Manual (16 page PDF file) add many capabilities of advanced graphing calculators to Sketchpad, including finding roots, plotting tangents, and calculating integrals.

Bjørn Felsager, 2004

### Piecewise Function Toolkit

This sketch contains tools and instructions for defining and graphing piecewise functions in Sketchpad. The tools can also be extended to graph functions with restricted domains.

Scott Steketee, 2008

### 3D Plotting Toolkit

This toolkit contains a template 3D coordinate system you can reuse in other sketches, and tools and instructions for plotting points (given 3D coordinates), segments, and curves.

Nick Jackiw, 2002

### 3D Invisibility Toolkit

The tools in this kit are designed for drawing dynamic models of convex polyhedra whose faces, edges, and vertices correctly change their appearance depending on their visibility: edges become dashed and faces and vertices vanish when they appear on the back side of the solid as its shape is changed by dragging vertices or by spinning the entire image. (Tools like these were used to construct the author's Icosahedron.)

This set of advanced analytic tools from Sketchpad's Samples collection includes calculations for tangents and function sectors, definitions of piecewise functions, and more.

### Boolean Tools

This sketch provides a set of Boolean operators, comparison tools, and Boolean functions, as well as detailed explanations of how these tools were created and how you can make them for yourself.

Scott Steketee, 2005

## 3D Explorations

Although Sketchpad's native domain is the geometry of the plane rather than the geometry of space, many sketches construct the projection of higher-dimensional mathematical models onto the two dimensions of the computer screen.

### Conic Connections

An extensive exploration of commonly studied properties of conic sections, including foci, directrices, eccentricities and discriminants, especially in the context of how these properties relate to the definition of conics as the intersection of a plane with a double-napped cone.

Kent Thele, May 2008

### 3D Gallery

A collection of constructions in three dimensions, including curves, surfaces, and various solids.

Nick Jackiw, Kate Mackrell, Paul Kunkel, Yao Liu.

### Icosahedron

A remarkable model of a regular icosahedron, with an exercise page describing a construction of this Platonic solid. The shape can be spun, lit from different angles, and viewed from any reference point. (Constructions like this can be made with the author's 3D Invisibility Toolkit.)

### 3D Toolkits

Check out the custom toolkits for developing 3D sketches available in the Toolkits section as well: 3D Plotting Toolkit; 3D Invisibility Toolkit.

## Beyond Euclid

Sketchpad is based largely on a Euclidean view of geometry. However, it is possible to both model other geometries within Euclidean geometry, and to use non-Euclidean capabilities in Sketchpad (such as the Calculator) to explore alternate geometries within Sketchpad. This section offers some examples.

### Minkowskian Geometry

This sketch and extensive documentation provide an introduction to the Minkowski geometry, instructions for creating constructions with the sketch provided, and notes for presenting a two-part Sketchpad workshop introducing Minkwoskian geometry.

Bjørn Felsager, 2004

### Poincaré Disk Model of Hyperbolic Geometry

This sketch from Sketchpad's Samples collection depicts the hyperbolic plane using the Poincaré Disk model, in which a line through two points is defined as the Euclidean arc passing through the points and perpendicular to the circle. Custom tools allow you to create constructions in the hyperbolic plane.

### Elliptical/Spherical Toolkit

Tools for exploring elliptical geometry, in which the Euclidean plane is replaced by the surface of a sphere, and lines in the plane are replaced by great circles on the sphere.

### Euler's Formula and the Complex Plane

A walk-through of the geometric interpretation of one of Euler's most famous results: eπi + 1 = 0.

Nick Jackiw, 2003

### Half Plane Model of Hyperbolic Geometry

Tools for constructing in the half-plane model of hyperbolic geometry, in which only points on one side of a horizontal boundary line are considered. A "line" through two points in this model is a semicircle whose center is on the boundary line. Tools for exploring isometries and triangle centers in this model are also included.

Judit Abardia Bochaca, 2004; Dan Bennett, 2001

## Machines and Mechanisms

Sketchpad provides an excellent medium for prototyping, visualizing, and demonstrating "mathematical machinery" of all forms. These sketches contain some ambitious mechanical constructions.

### Animated Billboard

This sketch simulates a changing billboard sign. The sign is made of vertical louvers, each of which is a triangular prism. By synchronizing the rotation of the louvers, three different sign faces alternate on the same billboard. The sketch includes a toolkit for making your own billboards, and extensive notes on the design and implementation of the simulation's sophisticated "conditional construction."

Paul Kunkel, 2007

This anthology collects several famous mechanical linkages—Peaucellier cell, a windshield wiper, and a sewing machine needle—in demonstrations both of their outer kinematics and their inner structure. An additional sketch on cam couplings allows you to design your own cam, either by cam shape or by desired rotary profile.

Nick Jackiw, 1994/2004

## Other Explorations

### Euler's Formula and the Complex Plane

A walk-through of the geometric interpretation of one of Euler's most famous results: eπi + 1 = 0.

Nick Jackiw, 2003

### Statistics Collection

This set of statistics visualizations and questions includes sketches of mean, least squares, r squared, the normal curve, comparing two means, and chi square.

Bill Finzer, 1993/2003

### Rainbow Investigation

This explanatory sketch from Sketchpad's Samples collection looks at the physics of rainbows: light traveling through prisms and water droplets.

### Dynamical Systems

This sketch from Sketchpad's Samples contains sample visualizations of several well-studied and potentially chaotic dynamical systems.

### Fractal Gallery

This sketch from Sketchpad's Samples illustrates a potpourri of iterated geometric constructions, many of them producing well-studied fractals.

### 360 Triangle Centers

This sketch constructs the first 360 Kimberling triangle centers of a single reference triangle, including its centroid, incenter, circumcenter, orthocenter—and several hundred more. Each center includes Kimberling index and barycentric coordinate equations.

Yao Liu, 2006