This learn node points to the page here at MIT Open Courseware for digital tools like the one illustrated above called Curves in Two Dimensions. There are more than two dozen tools for topics ranging such as precalculus, algebra and vectors, curves, surfaces and differential equations. In the MIT course with tools like the one shown are chapter outlines like this one called Curves about, as the Introduction explains:

“The tools of calculus developed so far allow us to describe most of the important properties of a smooth curve: which are its direction at any point, and how much it deviates from straightness there. This is measured by its curvature. How its path differs from planarity is measured by its torsion, also easily calculated.”

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This learn node looks first in the new issue of the math and arithmetic online magazine plus where this month an article titled What is the area of a circle? uses cake slices to illustrate some principles of the geometry of a circle. The author of the circle studies is Tom Ko�ner, a lecturer in the Department of Pure Mathematical Statistics at Cambridge University. The magazine brings math explanations from terrific teachers to online visitors. The circle at the right was the work of a student in an MIT class on Geometric Combinatorics. The right circle and formulas to explain it are on page 7 of the PDF called “Bounds of Crossing Numbers.”

Two stand-bys of open learning for math show up at the top of a Google search for “geometry circles.’” One is the colorful Cool Math that been nurtured for a decade, with some advertising support. The other is a Geometry Circles from Math for Morons Like Us, created by the ThinkQuest team in 1998. For more circle investigations, go to University of Birmingham Lecturer Chris Sangwin’s geoGebra page.

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This learn node features a video called “The Arch Never Sleeps” in which professors explain the mechanics of the support arches provide for structures. One professor points out the limitations of laying a block of stone across two others. The professor whose foot is shown as he stands on an arch (that is not glued together) is demonstrating the strength of stone arches. The video is on a page from the Open University Mathematics and Statistics modeling problems open courseware.

If the concepts of arches and mechanical forces get curiosity strongly aroused, a popular online set of notes for the mathematics of mechanics can be found at the University of Nebraska-Lincoln. Included are algebra, geometry, trigonometry, analytical geometry, calculus and vectors � as each of them relates to mechanics. Or for more concrete contemplations of arches mathematics and more, there is a page titled Geometry of Bridge Construction by a Jesuit teacher of math. That site includes a quick explanation of the famed seven Bridges of Konigsberg problem and Euler’s solution that provides a key basis for understanding how the connectivity of the Internet makes it possible for learn nodes to form the webs from which ideas can emerge. Related in time and math concepts are the Medieval breakthroughs in math visible in mosaics from Islamic buildings.

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This learn node cluster math help available online virtually from an amazing array of open sources. The picture here of Wolfgang Pauli and Niels Bohr as they “stare in wonder at a spinning top” is from lectures by David Tong of Cambridge University on Classical Dynamics. The picture is included in the third Tong lecture titled The Motion of Rigid Bodies. Pauli and Bohr � great mathematicians of the early 20th century � would surely turn the full intensity of their wonder on how a click of a 21st century mouse sends students to math help, math problems and math mentors.

In a click or two this learn node crosses the Atlantic pond from Cambridge to MIT for Algebra I lectures or to a place to think about geometry themes and variations while listening to some Bach.

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