nature's word : musings on sacred geometry

The Tetrahedron

The Tetrahedron at a Glance:

Number of faces:

4 triangles

Number of edges:

6

Number of vertices:

4

Dihedral angle:

70’32”

Facial angle:

60′

Central angle:

109’28”

Elemental attribution:

Fire

Geometric dual:

tetrahedron

 

Imaging the Tetrahedron:

small, basic animation (poorer quality, shorter download)
(IMAGE REMOVED)

larger, more complex animation (higher quality, longer download)
(IMAGE REMOVED)


ancient celtic model of the tetrahedron, carved in stone


an artist’s conceptualization of the tetrahedron


net, or pattern, that can be used to create a tetrahedron from cardstock

 

Proportions within the Tetrahedron

Proportions relative to edge length (if edge length equals one)

Insphere

Interspere

Circumsphere

Surface Area

0.204124145

0.353553391

0.612372436

 

 

square root of 2

divided by 4

 

 

Proportions relative to insphere (if insphere radius equals one)

Edge Length

Intersphere

Circumsphere

Surface Area

4.898979486

1.732050808

3

 

 

square root of 3

 

 

 

Proportions relative to intersphere (if intersphere radius equals one)

Edge Length

Insphere

Circumsphere

Surface Area

2.828427125

0.577350269

1.732050808

 

8 times (the

square root of 2

divided by 4)

1 divided by

the square root

of 3

square root of 3

 

 

Proportions relative to circumsphere

(if circumsphere radius equals one)

Edge Length

Insphere

Intersphere

Surface Area

1.632993162

0.33333333

0.577350269

 

 

1 divided by 3

1 divided by

the square root

of 3

 

 

Special thanks to Bruce
Rawles
for supplying the above listed proportional figures.


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3 Responses to “The Tetrahedron”

  1. neat stuff, cheers man

  2. Joof says:

    What aout the volumes of the platonic solids the their respective edge lengths. Do you have an equation for that? I am quite curious. Thank you for all the info.

  3. Aidrian O'Connor says:

    Joof – I worked out the above equations by looking at an awesome chart by Bruce Rawles, many many years ago. I just found it again – awesome that he still has it posted: http://www.geometrycode.com/free/polyhedra-math-tables-for-platonic-and-archimedean-solids/

    I worked the above out by making a spreadsheet that had many many permutations of multiplying the sqrt2, sqrt3, and phi by each other, dividing them, multiplying them, creating exponential values… and then I just methodically compared the results to Bruce’s table. It was pretty amazing to discover all the places that these values appeared.

    Bruce would probably be the guy to ask… he’s a mathematician… I’m a hack.

    Thanks for your comment – Aidrian