Our association with several members of the online community has been pivotal in getting us this far along. We want to give special thanks and credit to some of those individuals. Gratitude and credit go to Hans Milton for creating the mathematical model in Mathematica. Special thanks thanks to Strange for creating the graph using […]

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# 3D identities

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3D identities

# Tees and Mugs

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A t-shirt campaign at teespring.com/synchronous-geometry has been initiated in order to try and raise money. If enough interest is shown, we will be able to pay for either a mathematician to help us write a grant proposal, or to hire an animator to produce a short video that will better explain how the math works. In […]

# Contribute

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Currently, we are running this project as a hobby. Anyone wishing to participate in generating this material, or anyone who has any advice or expertise about what has already been accomplished, please register with the site and add your comments. If you can think of other ways to help, such as advice about how we […]

# Bizarro Vector

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There exists a deeper understanding of what constitutes -space. Each orthogonal two-dimensional plane is most commonly defined as two lengths (or axes) perpendicular to one another, sure, but what does that really mean? The concept of orthogonality brings the concept of relative direction (or orientation) into the picture. This structure of relative orientation, when combined […]

# How We Got Here

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200 years ago, Napier and others wrote about the Law of Sines for spherical angles, expanding on work begun by the Arabs. Approximately 150 years ago, a treatise on this branch of mathematics (Todhunter’s textbook “Spherical Trigonometry”) was published. From then until now, this was considered to be the complete story regarding spherical trigonometry. Nothing […]

# After Normalization

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Figure 4.3 Normalized Values for The curves representing have had the axis “normalized” and are rotated 90º to correspond to the orientation of the partial sine curve.

# Before Normalization

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Figure 4.2 Different Values of This is another graph by Strange and this one shows four different values: The curves have different extents along the x axis, which is the angle. In order to represent as a quantity that expresses direction, the curves must be “normalized” by making the extent of the curves […]

# Graphing the Function

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Figure 4.1 This graph was generated by Strange, using data from the Mathematica model written by Hans Milton.

# Variables

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C = circle made by intersection of the xy-plane and unit sphere S CO = (0,0,0) = origin = center of circle C r = = radius of circle C (r,0,0) = intersection of positive x-axis with unit sphere S = SN (north pole) (-r,0,0) = intersection of negative x-axis with unit sphere S […]