If you take a piece of card, mark out a pentagon, cut and remove one
of the five triangular 'sections' and join the two cut edges together,
the resultant 3D shape (we called it a 'squone") seems to create an
angle at it's 3D-"tip" that is also 108deg, ie the amount of 'angle'
removed from the perimeter. Is there a name for this 3D shape, and
does it belong to a family of shapes that behave in a similar way? I
can mail a diagram.

---

Request for Question Clarification by nenna-ga on 18 Sep 2002 07:53 PDT

We can not give out our mail addresses, per Google policy, but, you're
free to take pictures and post them on a site with the URL here so
that we may look at them.
Many sites do free web hosting, such as Fotki.com

Nenna-GA

---

Request for Question Clarification by rbnn-ga on 18 Sep 2002 10:16 PDT

I don't quite understand what the shape is that you are creating. By
"a piece of card" do you mean " a piece of card paper" or "a playing
card"? By the phrase "mark out a pentagon" do you mean just draw a
pentagon (I assume) or actually cut out the pentagon?

Once I draw the (regular?) polygon on the card, where are the five
triangular sections?

Perhaps if you just described the planar coordinates of the vertices
of the polygons you are creating that would help.

Hi isambardbuckminster-ga, 

What you have created is known as a SQUARE PYRAMID – a pentahedron
(5-sided 3-d object) with a square base and a pyramid made of 4
triangular “walls.”  Pyramids are named for the shape of the polygon
which makes up their base, hence the term “square.”

Assuming we started with a perfect pentagon, our triangles will be
equilateral and therefore the shape is further defined as a “regular
polyhedron” known as a Johnson Solid.

There are 92 Johnson Solids – defined as convex polyhedra with regular
faces and equal edge lengths. 28 of these, including the square
pyramid, are considered to be simple regular-faced polyhedra - That is
they cannot be dissected into two other regular-faced polyhedra by a
plane.

The second part of your question doesn’t really apply to 3-d shapes.
The angle at the “tip” of your pyramid is not measured in degrees
because it is a 3-dimensional shape and degrees are a 2-dimensional
measurement. To refer to degrees here would be equivalent to asking
“how many feet are in an acre?”  Instead a 3-d unit must be used. This
unit is called "steradian." It is the measure of a solid angle.
Steradians relate solid
angles (which you can imagine as cones or triangles radiating out from
a point) to angles that subtend an entire sphere (4*Pi).

See the article below: “Ask Dr. Math – Measuring angles using
steradians” to see the mathematics involved in the calculation of
steradians.



=====================================
FURTHER EXPLANATION
=====================================


MATHWORLD – THE SQUARE PYRAMID

http://mathworld.wolfram.com/SquarePyramid.html



MATHWORLD – JOHNSON SOLID

http://mathworld.wolfram.com/JohnsonSolid.html


ASK DR. MATH – DEGREES IN A SPHERE?

“What is a solid angle? One way to picture a solid angle is the tip of
a cone or a pyramid. A tall narrow cone has a small solid angle at the
tip; a broad flat cone has a large solid angle at the tip.”

http://mathforum.org/library/drmath/view/55358.html



ASK DR. MATH – MEASURING ANGLES USING STERADIANS

http://mathforum.org/library/drmath/view/54799.html



ASK DR. MATH – USE OF STERADIANS 

http://mathforum.org/library/drmath/view/51707.html


So that should answer your question about your “mystery” shape.
Although you formed it in a non-conventional way, the resultant 3-d
figure is indeed a square pyramid.

Thanks for a great question! 

--K~

Search terms:

Pyramid geometry
“Square pyramid” geometry
Pyramid pentagon geometry
Pentahedron
Pyramid angle measurement
Steradian pyramid

---

Request for Answer Clarification by isambardbuckminster-ga on 02 Oct 2002 03:12 PDT

I have uploaded the diagram requested to
http://home.clara.net/alanperks/Image01001.jpg

Sorry, it's definitely not a square pyramid. As you will see, the
produced 3D shape has no 'edges' except the 4 arcs at the 'base', but
a continuous curved surface. The 'base' is 'open'. It is a two-sided
object, not 5 faces. I'm sorry everybody has misunderstood. Try to
make one, it is beautiful.

Thanks anyway for the stuff about steradians, all new to me. However,
my concern (if the shape is recognised) is whether the angle at the
produced tip relates in any way to the angle(s) at the (flat) corners
(they look the same) and if so is there some kind of rule that
predicts other members of the family?

Really hope you can follow this up

IB-

---

Clarification of Answer by knowledge_seeker-ga on 19 Oct 2002 12:22 PDT

Hi again –

Well, I didn’t give up on this question, but I can honestly say I find
nothing that discusses the object you have created.

The difficulty is that the curved surface of the shape you define is a
function of the property of the material used to create the object,
not of the object itself. In other words, because you have made it of
paper (or cardboard) you get that nice cone shape and those arced
edges. (It IS a beautiful shape by the way!).

Now, once you get a cone shape you really are talking about a
3-dimensional object -– bottom or no bottom --geometry assumes a
closed (solid) object.  Of course, it isn’t really a cone because the
base has “corners” whereas a cone is defined as a pyramid with a
circular cross section.

To explain a different way, what you’ve done is “morphed” a flat
polygon into a curved 3-d object, and this is only possible because
you used a flexible medium to build it.

Given this, I cannot find a way to apply regular mathematical formulas
to the shape in order to find a rule that relates the angle at the tip
to the angles of the flat corners.

However, all that said, I am certainly no expert in solid or plane
geometry. (I’m an expert in finding things, not necessarily doing
them!)  If humans (or computers)can mathematically predict airplane
turbulence, I am sure there must be ways to predict the angles of your
object!

Maybe the answer is to ask your question to the math experts at one of
the sites I gave you above. Ask Dr. Math seems to be the reference of
choice for students.  Make sure when you do it, you include the link
to your picture and a link to this Google Question so the expert can
see what ground we’ve already covered.

Or try the math department of your local university. Maybe a dialogue
back and forth with a professor will get you the answers you want.

Good luck with your quest!  Now, I think what I’m going to do is make
a bunch of these shapes out of little bits of silver and gold origami
paper and have the kids decorate them for Christmas ornaments.

Thanks for sharing your discovery –

-K~

The paid answer was good, but the second comment is perfect as it
explains the maths process to arrive at the definitive answer. Many
many many thanks to everyone.

Is this not simply a square-based pyramid? Or have I misunderstood something ... ?

Regards
iaint-ga

Here are some comments which may answer your question more fully:

1. The shape you describe is basically a cone which has been cut off
along 4 curves, and these curves are "geodesics" in the conical
geometry (this means that if you lived on the cone you would think the
curves were completely straight).  These curves all have the same
length, and they form a 'square' which has center equal to the cone
vertex.  In your particular case, the distance from the vertex of the
cone to any of the vertices of the square (let's call this r, the
radius) is fixed by the geometry of the pentagon (the distance is a
little shorter than the length of each edge of the square (call this
distance l).

2. This shape fits into a general class of cone - based shapes; to
construct more of them, simply take any regular polygon with n sides
and remove some number of sides, say k.  Then glue as you have
described to form a cone.

The question becomes: What is the cone angle? Call "a" the cone angle,
i.e. the angle from the line of symmetry of the cone to the surface.

Then it is easy to see that sin(a)=(1 - k/n).

The angle visible in your diagram (the projection of the cone) is
actually 2a, and the vertex angle of a pentagon is 3pi/5,

so for a pentagon with one triangle removed we obtain:

projection cone angle (2a):  1.855 radians
vertex angle (pentagon):  1.888 radians

As you can see, these angles are pretty close! But they aren't equal,
good observation though!