Hi Adrian,

> It looks like trapezohedron is the winner by a long shot! anti-dipyramid seems
like a purely academic term. But I ended up

I just found this. To complicate the naming even more, the particular type of
trapezohedron the script makes is called a "Streptohedron".

In the script, a pyramid it turned half way in respect the other. But consider
keeping one still and rotating the other one, the faces can go through a series
of trapazoidal shapes.

Here is the Wolfram demostration of this. "watch web preview" will go through
the applet features.

http://demonstrations.wolfram.com/StreptohedronAndTrapezohedron/

When it is approached like this, the center antiprism changes height. So the
script won't do it as is and the trapazoids will have a crease in them.

Roger

Hi Adrian,

> In the script, a pyramid it turned half way in respect the other. But consider
keeping one still and rotating the other one, the faces can go through a series
of trapazoidal shapes.

I added a little to the script to see if rotations between 0 and 90 degrees
could generate a model. To do that, I multipled the height of the antiprism by
the sin() of a fourth argument being the rotation angle.

The script doesn't generate the antiprism but rather just takes the convex hull
of the to pyramids. The output model includes the two pyramids and the convex
hull for comparison.

At 0 and 90 degrees it does the correct thing. But at the range in between it is
almost right but not quite. Tantalizingly close! Script below.

Roger

#!/usr/bin/python

import os
import sys
import math

prog = "pyr2delt.py"
if(len(sys.argv) != 4):
     sys.stderr.write(prog + ": wrong number of arguments\nusage: "+prog+"
pyr_poly_fraction height rot_angle\n")
     sys.exit(1)

sides_txt = sys.argv[1].split("/")
if(len(sides_txt)==1):
     sides_txt.append("1")

sides = []
for txt in sides_txt:
     try:
        sides.append(int(txt))
     except:
        sys.stderr.write(prog + ": number of sides: '"+sys.argv[1]+"' is not an
integer\n")
        sys.exit(1)

if(sides[0] < 3):
     sys.stderr.write(prog + ": number of sides: '"+sides[0]+"' is not 3 or
greater\n")
     sys.exit(1)
if(sides[1] < 1):
     sys.stderr.write(prog + ": denominator of sides: '"+sides[1]+"' is not 1 or
greater\n")
     sys.exit(1)
if(float(sides[0])/sides[1] <= 0 ):
     sys.stderr.write(prog + ": sides: denominator is greater than the number of
sides '"+sys.argv[1]+"'\n")
     sys.exit(1)

try:
     pyr_ht = float(sys.argv[2])
except:
     sys.stderr.write(prog + ": height: '"+sys.argv[2]+"' is not a number\n")
     sys.exit(1)

try:
     rot_angle = float(sys.argv[3])
except:
     sys.stderr.write(prog + ": height: '"+sys.argv[3]+"' is not a number\n")
     sys.exit(1)

ang = math.pi*sides[1]/sides[0]

corrector = math.sin((math.pi/180.0)*rot_angle)

ant_ht = pyr_ht*(1/math.cos(ang)-1)*corrector
total_ht = ant_ht+(pyr_ht*2)

sys.stderr.write("pyramid   = "+str(pyr_ht)+"\n")
sys.stderr.write("antiprism = "+str(ant_ht)+"\n")
sys.stderr.write("total     = "+str(total_ht)+"\n")

files = " tmp[1-3].off"
os.system("rm" + files)
os.system("polygon -t pyr %d/%d -l %.17f | off_trans -R 0,0,%.17f -T 0,0,%.17f >
tmp1.off" % (sides[0], sides[1], pyr_ht, ang*rot_angle/math.pi, ant_ht/2))
os.system("polygon -t pyr %d/%d -l %.17f | off_trans -M 0,0,1 -R 0,0,-%.17f -T
0,0,-%.17f > tmp2.off" % (sides[0], sides[1], pyr_ht, ang*rot_angle/math.pi,
ant_ht/2))
os.system("off_util tmp1.off tmp2.off | conv_hull > tmp3.off")
os.system("off_util" + files)

Hi Roger

On Mon, 5 Jul 2010, Roger Kaufman wrote:
> http://en.wikipedia.org/wiki/Truncated_trapezohedron

These look interesting.

The square model on that page appears to have more than
one edge length. There may be a unique solution (for each
polygon) having a single edge length (possibly with a
unique second solution if considering a "crossed" version
comes from the same polygon).


> I am convinced the script is the ideal way to build a custom
> trapezohedron. It only requires the height of the pyramidal portion. If

It is clear. Specifying the antiprism "belt" would be an
alternative.

I was thinking that there will be a particular size of antiprism
whose dual has the same antiprism for a belt. e.g.

     polygon -t ant  5 -E 0.7435 | pol_recip -a -r E | antiview

I'll look at a calculation for that.

It is interesting to compare this with a self-dual trapezohedron
with one apex truncated. I have included one below. I am not
sure if it is possible to be exctly self-dual, as per above
(it seems like it could be possible)

     pol_recip -c 0,0,0 -r 1,8 -a aug_anti.off | antiview

The canonical version is self-dual

     canonical aug_anti.off | pol_recip -c 0,0,0 -a | antiview


> It is not a great polyhedra to generate custom sizes, but it seem like
> having a kite cross member of size 1, and the pyramid height would be
> the most orderly way to work with them.

In the polygon program, the kite cross would have to be the polygon
edge. There could be an option to produce a trapezohedron if a
pyramid or an antiprism were specified.


> http://mathworld.wolfram.com/Trapezohedron.html
>
> The calculations for various dimensions are not very nice! They show the
> relative values for duals of regular antiprisms and the formula are very
> complex.

The page seems to be defining a trapezohedron as a dual of
an Archimedian antiprism. The Wikipedia page allows for
non-Archimediean antiprisms.

The MathWorld values might be tidier if they were expressed in
terms of angles.

Adrian.
--
Adrian Rossiter
adrian@...
http://antiprism.com/adrian


OFF
11 11 0
-0.5000000000000003 0.6881909602355868 0.2628627764032445
0.8090169943749476 -0.2628655560595669 0.2628627764032446
0.4999999999999999 0.688190960235587 0.2628627764032446
-0.5000000000000001 -0.6881909602355872 -0.2628627764032443
-0.8090169943749477 0.2628655560595669 -0.2628627764032446
-0.8090169943749477 -0.262865556059567 0.2628627764032447
0.5 -0.688190960235587 -0.2628627764032445
-6.938893903907228e-17 -0.8506508083520402 0.2628627764032446
0.8090169943749476 0.262865556059567 -0.2628627764032447
-2.582713617087596e-16 0.8506508083520401 -0.2628627764032446
-6.938893903907228e-17 -3.330669073875469e-17 2.489871955600187
4 2 9 0 10
3 4 0 9
4 0 10 5 4
4 6 1 10 7
3 8 6 1
3 8 2 9
4 10 1 8 2
3 3 4 5
3 3 7 6
5 8 6 3 4 9
4 10 5 3 7

Hi Adrian:
      I can run antiview with my DOS window when I am in the bin
folder.
      I am not able to get it running from a higher directory.
      This is my path.

C:\Python25; C:\antiprism-0.19.2\bin; C:\Program Files\Image Magick-6.6.0-Q16

       Can you see anything wrong? Python runs just fine from the
desktop directory when I first start up the DOS window. This is for
my Vista laptop.
       It is same for my desktop which has windows XP. The path does
not seem to work there either.
       So the Path works for Python and not antiprism.


dharmraj

Hi Roger

On Mon, 5 Jul 2010, Roger Kaufman wrote:
> I just found this. To complicate the naming even more, the particular
> type of trapezohedron the script makes is called a "Streptohedron".
>
> In the script, a pyramid it turned half way in respect the other. But
> consider keeping one still and rotating the other one, the faces can go
> through a series of trapazoidal shapes.

Searching on Streptohedron it seems that it is a usually a name for
a polyhedron produced by the n_icon process. The MathWorld applet
doesn't seem to involve this process.


> http://demonstrations.wolfram.com/StreptohedronAndTrapezohedron/
>
> When it is approached like this, the center antiprism changes height. So
> the script won't do it as is and the trapazoids will have a crease in
> them.

I was thinking before of adding an option to 'polygon' to have the
antiprism polygons at different angles. The duals of these would
have a fixed height for a fixed height antiprism. I didn't add the
option in as I didn't think the polyhedra would be of interest.
However, maybe they would be. Also, they are examples of polyhedra
with full symmetry 'Dn', and so useful to have around for that.

Adrian.
--
Adrian Rossiter
adrian@...
http://antiprism.com/adrian

Hi dharmraj,

>      I am not able to get it running from a higher directory.
>      This is my path.
>
> C:\Python25; C:\antiprism-0.19.2\bin; C:\Program Files\Image Magick-6.6.0-Q16

I cut and pasted your path in a dos window as

set PATH=text from above...

It fails because of the space before C:

If I tried it with quotes as

set PATH="text from above..."

It put the quotes themselves into the PATH variable!

I am thinking the spaces are what is confusing it.

Roger

Hi dharmraj

On Tue, 6 Jul 2010, swdharmraj wrote:
> C:\Python25; C:\antiprism-0.19.2\bin; C:\Program Files\Image Magick-6.6.0-Q16
>
>      Can you see anything wrong? Python runs just fine from the
> desktop directory when I first start up the DOS window. This is for

The spaces after the semicolons may be causing a problem. If you
delete those then the Antiprism path might work. Also, the
ImageMagick path contains spaces internally. They probably need
escaping, or you could probably just put quotes around the whole
ImageMagick path.

Adrian.
--
Adrian Rossiter
adrian@...
http://antiprism.com/adrian

> The spaces after the semicolons may be causing a problem. If you
> delete those then the Antiprism path might work. Also, the
> ImageMagick path contains spaces internally. They probably need
> escaping, or you could probably just put quotes around the whole
> ImageMagick path.
>
> Adrian.
> --



Thank you. That did it.

dharmraj

Hi Adrian,

This will be a composite reply as I have made many comments.

> Searching on Streptohedron it seems that it is a usually a name for
> a polyhedron produced by the n_icon process. The MathWorld applet
> doesn't seem to involve this process.
>
>
> > http://demonstrations.wolfram.com/StreptohedronAndTrapezohedron/

The word is hardly used anywhere. I think it might be used incorrectly. I will
bring this to Sandor's attention.


> > When it is approached like this, the center antiprism changes height. So
> > the script won't do it as is and the trapazoids will have a crease in
> > them.
>
> I was thinking before of adding an option to 'polygon' to have the
> antiprism polygons at different angles. The duals of these would
> have a fixed height for a fixed height antiprism. I didn't add the
> option in as I didn't think the polyhedra would be of interest.
> However, maybe they would be. Also, they are examples of polyhedra
> with full symmetry 'Dn', and so useful to have around for that.

I did a binary search to find the value to make the script I last sent work in
just one custom case. It adds a minute about to the corrector as in

corrector = math.sin((math.pi/180.0)*rot_angle) + 0.0330016863393

Then it produces the trapezohedron for which I included the OFF file below.

pyr2test.py 4 0.70710678118654752440084436210485 45 | conv_hull

In this case there is no argument on behalf of the United States that these are
trapazoids! The dual is just an antiprism which is not fully twisted top and
bottom and the height is changed. However, the dual is probably not properly an
antiprism either and who knows what to call it!

The problem with the script modification will be that n/m versions can't be
created this way.

It might be if the same kind of twist could be done on the antiprism, but now
the polygon program but then it would become more complex.

Just a note, I was at some point going to work on a de-planarizer. Now there is
the ideal situation for it.


>I was thinking that there will be a particular size of antiprism
>whose dual has the same antiprism for a belt. e.g.

The trapazohedron can be thought of as a stellated antiprism. I, too, was
wondering if there were spacial cases such that the antiprism and its dual could
be superimposed. This would make another nice blended model.

Roger

OFF
10 8 0
-0.1379496896414716 -0.1083863756623685 0.6935199226610738
0.6935199226610738 -0.1083863756623685 0.1379496896414715
0.1379496896414714 0.1083863756623685 0.6935199226610738
-0.1379496896414714 0.1083863756623685 -0.6935199226610738
0 0.8154931568489161 -4.993290506879256e-017
-0.6935199226610738 -0.1083863756623685 -0.1379496896414716
0.1379496896414716 -0.1083863756623685 -0.6935199226610738
0 -0.8154931568489161 4.993290506879256e-017
0.6935199226610738 0.1083863756623685 -0.1379496896414715
-0.6935199226610738 0.1083863756623685 0.1379496896414714
4 0 9 5 7
4 7 1 2 0
4 9 0 2 4
4 1 8 4 2
4 6 8 1 7
4 7 5 3 6
4 6 3 4 8
4 3 5 9 4

http://mathworld.wolfram.com/Cube-OctahedronCompound.html
http://en.wikipedia.org/wiki/Compound_of_cube_and_octahedron

http://mathworld.wolfram.com/StellaOctangula.html
http://en.wikipedia.org/wiki/Stellated_octahedron

--- In antiprism@yahoogroups.com, "Roger Kaufman" <vortexswirling@...> wrote:
>
> The trapazohedron can be thought of as a stellated antiprism. I, too, was
wondering if there were spacial cases such that the antiprism and its dual could
be superimposed. This would make another nice blended model.

Hi Alan,

> http://mathworld.wolfram.com/Cube-OctahedronCompound.html

I was referring to what I am calling the top and bottom stellation of an
antiprism. Are there cases such that the dual of the antiprism equals that
stellation.

Adrian had a near example and with a few more decimal places it could be done.
The length of the diagonal edges of the antiprism belt are 0.743496068920667

Here are models showing the antiprism opposed and then rotated into position. It
shows the antiprism is coplanar with its dual.

http://www.interocitors.com/tmp/trap_and_dual_0_deg.wrl

http://www.interocitors.com/tmp/trap_and_dual_36_deg.wrl

The height of the antiprism is 0.52573111212

The commands I used to make this were

pyr2delt.py 5 2.227032728825 | conv_hull | off_color -f red -o tmp1b.off
pol_recip tmp1b.off | off_trans -R 0,0,36 | off_color -f yellow -e none -o
tmp2b.off
off_util tmp1b.off tmp2b.off | off_trans -R 90,0,0 -o tmp3b.off
#off_query El -I 0,15 -e 0,15 tmp3b.off
off_color -e lightgrey -v gold tmp3b.off

I was using off_query to output the height of the antiprism and used this to
zero in on the right height.

Roger

That sounds as if you are stellating the top and bottom of an octahedron until
it becomes a hexahedron.
See the
http://groups.yahoo.com/group/synergeo/messages/15504?threaded=1&m=e&tidx=1

--- In antiprism@yahoogroups.com, "Roger Kaufman" <vortexswirling@...> wrote:
>
> Hi Alan,
>
> > http://mathworld.wolfram.com/Cube-OctahedronCompound.html
>
> I was referring to what I am calling the top and bottom stellation of an
antiprism. Are there cases such that the dual of the antiprism equals that
stellation.
>
> Adrian had a near example and with a few more decimal places it could be done.
The length of the diagonal edges of the antiprism belt are 0.743496068920667
>
> Here are models showing the antiprism opposed and then rotated into position.
It shows the antiprism is coplanar with its dual.
>
> http://www.interocitors.com/tmp/trap_and_dual_0_deg.wrl
>
> http://www.interocitors.com/tmp/trap_and_dual_36_deg.wrl
>
> The height of the antiprism is 0.52573111212
>
> The commands I used to make this were
>
> pyr2delt.py 5 2.227032728825 | conv_hull | off_color -f red -o tmp1b.off
> pol_recip tmp1b.off | off_trans -R 0,0,36 | off_color -f yellow -e none -o
tmp2b.off
> off_util tmp1b.off tmp2b.off | off_trans -R 90,0,0 -o tmp3b.off
> #off_query El -I 0,15 -e 0,15 tmp3b.off
> off_color -e lightgrey -v gold tmp3b.off
>
> I was using off_query to output the height of the antiprism and used this to
zero in on the right height.
>
> Roger
>

Hi Alan,

> That sounds as if you are stellating the top and bottom of an octahedron until
it becomes a hexahedron.
> See the
http://groups.yahoo.com/group/synergeo/messages/15504?threaded=1&m=e&tidx=1

That can be done when the octahedron is set to the height of sqrt(1/3). Then the
pyramids top and bottom are also height sqrt(1/3).

The octahedron (now really should be called a triangular antiprism), has unit
edges for the top and bottom triangles. The diagonal edge lengths are sqrt(1/6).

The dual is a rhombohedron. In this model the vertical length is sqrt(3). The
angle of the rhombohedron is 75.5225. The edge lengths are sqrt(1/6).

http://www.interocitors.com/tmp/trap_and_dual_octahedron.wrl

Roger

Hi Roger

On Tue, 6 Jul 2010, Roger Kaufman wrote:
>> I was thinking before of adding an option to 'polygon' to have the
>> antiprism polygons at different angles. The duals of these would
>> have a fixed height for a fixed height antiprism. I didn't add the
>> option in as I didn't think the polyhedra would be of interest.
>> However, maybe they would be. Also, they are examples of polyhedra
>> with full symmetry 'Dn', and so useful to have around for that.
>
> I did a binary search to find the value to make the script I last sent
> work in just one custom case. It adds a minute about to the corrector as
> in

There will be an expression for the height. It might not be
too bad. Even if it is, the height can still be found using
the nearest point on the plane of the antiprism's side faces.


> In this case there is no argument on behalf of the United States that
> these are trapazoids! The dual is just an antiprism which is not fully
> twisted top and bottom and the height is changed. However, the dual is
> probably not properly an antiprism either and who knows what to call it!

I don't know what those are called, maybe twisted right antiprisms?


> It might be if the same kind of twist could be done on the antiprism,
> but now the polygon program but then it would become more complex.

I have added it in (I haven't pushed the changes yet). It only
currently works with antiprisms. The height is that of an
antiprism with the given parameters, then the twist is applied.


>> I was thinking that there will be a particular size of antiprism
>> whose dual has the same antiprism for a belt. e.g.
>
> The trapazohedron can be thought of as a stellated antiprism. I, too,
> was wondering if there were spacial cases such that the antiprism and
> its dual could be superimposed. This would make another nice blended
> model.

After a few calculations the result turns out to be quite
simple. If the polygon circumradius is R then the antiprism
height is

     R*sqrt((1-cos(m*pi/n))*2)

And the reciprocation sphere radius is

     R*sqrt((cos(m*pi/n)+1)/2)

I have included a script below

Adrian.
--
Adrian Rossiter
adrian@...
http://antiprism.com/adrian

#!/usr/bin/python

import os
import sys
import math

prog = "ant2pr.py"
if(len(sys.argv) != 2):
     sys.stderr.write(prog + ": wrong number of arguments\nusage: "+prog+"
pyr_poly_fraction\n")
     sys.exit(1)

sides_txt = sys.argv[1].split("/")
if(len(sides_txt)==1):
     sides_txt.append("1")

sides = []
for txt in sides_txt:
     try:
        sides.append(int(txt))
     except:
        sys.stderr.write(prog + ": number of sides: '"+sys.argv[1]+"' is not an
integer\n")
        sys.exit(1)

if(sides[0] < 3):
     sys.stderr.write(prog + ": number of sides: '"+sides[0]+"' is not 3 or
greater\n")
     sys.exit(1)
if(sides[1] < 1):
     sys.stderr.write(prog + ": denominator of sides: '"+sides[1]+"' is not 1 or
greater\n")
     sys.exit(1)
if(float(sides[0])/sides[1] <= 0 ):
     sys.stderr.write(prog + ": sides: denominator is greater than the number of
sides '"+sys.argv[1]+"'\n")
     sys.exit(1)


ht = 3.0
ant_ht = ht;

ang = math.pi*sides[1]/sides[0]
cos_a = math.cos(ang)
R = 1/(2*math.sin(ang))

ant_ht    = R*math.sqrt((1-cos_a)*2)
recip_rad = R*math.sqrt((cos_a+1)/2)

files = " tmp1.off"
os.system("rm" + files)
os.system("polygon -t ant %d/%d -l %.17f > tmp1.off" % (sides[0], sides[1],
ant_ht))
os.system("pol_recip -a -r %.17f tmp1.off" % (recip_rad))

Hi Roger

On Wed, 7 Jul 2010, Adrian Rossiter wrote:
>>> I was thinking that there will be a particular size of antiprism
>>> whose dual has the same antiprism for a belt. e.g.
...
> After a few calculations the result turns out to be quite
> simple. If the polygon circumradius is R then the antiprism
...
> I have included a script below

Looking at this. It appears that for {2n/2m} the model
shows (visibly only) a pair of self-dual singly truncated
{n/m} trapezohedra.

In which case it would seem that the canonical version isn't
the only form that is self-dual by reciprocation in a sphere.

Adrian.
--
Adrian Rossiter
adrian@...
http://antiprism.com/adrian

On Wed, 7 Jul 2010, Adrian Rossiter wrote:
> Looking at this. It appears that for {2n/2m} the model
> shows (visibly only) a pair of self-dual singly truncated
> {n/m} trapezohedra.
>
> In which case it would seem that the canonical version isn't
> the only form that is self-dual by reciprocation in a sphere.

Thinking about it, it ay well be that any singly
truncated trapezohedron is self-dual by reciprocation
in an appropriate sphere, and some may be self-dual
in two spheres.

Adrian.
--
Adrian Rossiter
adrian@...
http://antiprism.com/adrian

Hi Adrian,

> Looking at this. It appears that for {2n/2m} the model
> shows (visibly only) a pair of self-dual singly truncated
> {n/m} trapezohedra.
>
> In which case it would seem that the canonical version isn't
> the only form that is self-dual by reciprocation in a sphere.

I am not sure what you mean. I tried 10/4 and it just looks like two 5/2.

But did you try 5/3? The dual comes out the same height and appears to have the
same triangles and can superimposed, but on a closer look they consist of darts!
Quite nice.

Roger

Hi Adrian,

> > work in just one custom case. It adds a minute about to the corrector as
> > in
>
> There will be an expression for the height. It might not be
> too bad. Even if it is, the height can still be found using
> the nearest point on the plane of the antiprism's side faces.

In any case, the script can produce one from a pyramid.

But what I was referring to was the possibility of producing ones that are not
made of kites. In the last script I sent I had added a corrector for the
antiprism height but it is just a little off.

Or are you taking this into account in the polygon program?

Roger

Hi Roger

On Wed, 7 Jul 2010, Roger Kaufman wrote:
>> Looking at this. It appears that for {2n/2m} the model
>> shows (visibly only) a pair of self-dual singly truncated
>> {n/m} trapezohedra.
>>
>> In which case it would seem that the canonical version isn't
>> the only form that is self-dual by reciprocation in a sphere.
>
> I am not sure what you mean. I tried 10/4 and it just looks like two
> 5/2.

That is true! The model I had in mind has one of the pyramids
removed at each end. There is also an antiprism base polygon
to remove with these, so it is the faces of two complete pyramids
that are removed.

Adrian.
--
Adrian Rossiter
adrian@...
http://antiprism.com/adrian

Hi Roger

On Wed, 7 Jul 2010, Roger Kaufman wrote:
>>> work in just one custom case. It adds a minute about to the corrector as
>>> in
>>
>> There will be an expression for the height. It might not be
>> too bad. Even if it is, the height can still be found using
>> the nearest point on the plane of the antiprism's side faces.
>
> In any case, the script can produce one from a pyramid.
>
> But what I was referring to was the possibility of producing ones that
> are not made of kites. In the last script I sent I had added a corrector
> for the antiprism height but it is just a little off.
>
> Or are you taking this into account in the polygon program?

I was thinking of the twisted antiprism versions. I am adding
the trapezohedra into polygon. I have already added the
trapezohedra from right antiprisms and pyramids.

I am looking at the calculations for the twisted version now. I
am not sure yet whether the expression for the height will be
simplify.

Adrian.
--
Adrian Rossiter
adrian@...
http://antiprism.com/adrian

On Wed, 7 Jul 2010, Adrian Rossiter wrote:
> removed at each end. There is also an antiprism base polygon
> to remove with these, so it is the faces of two complete pyramids
> that are removed.

That isn't correct in the way I meant it. The base polygons
that are removed are the ones that are turned a half-edge
from the pyramid base polygons.

I should probably just post a model!

Adrian.
--
Adrian Rossiter
adrian@...
http://antiprism.com/adrian

On Wed, 7 Jul 2010, Adrian Rossiter wrote:
> I should probably just post a model!

Here is a model showing a base trapezohedron, the compound
of two of them, and the dual pair of the self-dual singly
truncated trapezohedron

     http://www.antiprism.com/misc/gyro_pyr_pr.wrl

Hopefully that is clearer than my attempts to describe
this in words!

Adrian.
--
Adrian Rossiter
adrian@...
http://antiprism.com/adrian

Hi Roger

On Wed, 7 Jul 2010, Adrian Rossiter wrote:
> I was thinking of the twisted antiprism versions. I am adding
> the trapezohedra into polygon. I have already added the
> trapezohedra from right antiprisms and pyramids.
>
> I am looking at the calculations for the twisted version now. I
> am not sure yet whether the expression for the height will be
> simplify.

The final expression is fairly tidy.

An {m/n} antiprism has height H and its polygons twisted
towards each other each by an angle of 'b'. The height of
the corresponding trapezohedran is

     H * (cos(pi*m/n) + cos(b)) / (cos(pi*m/n) - cos(b))

I slogged through a lot of trig to get this, but there is
likely some relationship or property that leads more directly
to the solution.

I'll do some more testing and if all is well I'll push
the changes a bit later.

Adrian.
--
Adrian Rossiter
adrian@...
http://antiprism.com/adrian

Hi Adrian,

> Here is a model showing a base trapezohedron, the compound
> of two of them, and the dual pair of the self-dual singly
> truncated trapezohedron
>
>     http://www.antiprism.com/misc/gyro_pyr_pr.wrl

So it looks like you are saying that there is an antiprism height such that if
stellated on just one side (the base) it is self dual.

Then if stellated on both sides the height is the same as the base + the dual
compound.

Roger

Hi Adrian,

>     H * (cos(pi*m/n) + cos(b)) / (cos(pi*m/n) - cos(b))

It looks like this is keeping the height of the antiprism constant.

A nice animation would be an octahedron, thought of as two J1's with a zero
height antiprism. It would twist causing the antiprism to rise up between them
until it became the trapezohedron of the J1 pyramid.

This was what I was attempting to do in the script. I am sure there is a formula
for the antiprism height through this motion.

In the polygon program it looks like the -t parameter already acts like
get_arg_id()

Roger

Hi Roger

On Wed, 7 Jul 2010, Roger Kaufman wrote:
>>     http://www.antiprism.com/misc/gyro_pyr_pr.wrl
>
> So it looks like you are saying that there is an antiprism height such
> that if stellated on just one side (the base) it is self dual.
>
> Then if stellated on both sides the height is the same as the base + the
> dual compound.

I originally thought this particular one side stellation would
be special in being self-dual, but I now think that they will
generally be self-dual and so it is only special in that in
this self-duality it reciprocates the apex onto the plane of
the middle band of vertices.

The compound {2n/2m} special height antiprism and trapezohedron
pair is then like a compound of two of the special {n/m} self-dual
pairs (a compound of two of the models in the right of the
VRML.)

Adrian.
--
Adrian Rossiter
adrian@...
http://antiprism.com/adrian

Hi Adrian,

> The compound {2n/2m} special height antiprism and trapezohedron
> pair is then like a compound of two of the special {n/m} self-dual
> pairs (a compound of two of the models in the right of the
> VRML.)

Do you have that special height in terms of the pyramid height?

Roger

Hi Roger

On Wed, 7 Jul 2010, Roger Kaufman wrote:
> Do you have that special height in terms of the pyramid height?

I don't have a very good expression for it, but it could be e.g.

     R * sqrt((1+cos(pi*m/n)*2) / tan(pi*m/n)

I have included a script below.

Adrian.
--
Adrian Rossiter
adrian@...
http://antiprism.com/adrian

#!/usr/bin/python

import os
import sys
import math

prog = "pyr2trap_pr.py"
if(len(sys.argv) != 2):
     sys.stderr.write(prog + ": wrong number of arguments\nusage: "+prog+"
pyr_poly_fraction\n")
     sys.exit(1)

sides_txt = sys.argv[1].split("/")
if(len(sides_txt)==1):
     sides_txt.append("1")

sides = []
for txt in sides_txt:
     try:
        sides.append(int(txt))
     except:
        sys.stderr.write(prog + ": number of sides: '"+sys.argv[1]+"' is not an
integer\n")
        sys.exit(1)

if(sides[0] < 3):
     sys.stderr.write(prog + ": number of sides: '"+sides[0]+"' is not 3 or
greater\n")
     sys.exit(1)
if(sides[1] < 1):
     sys.stderr.write(prog + ": denominator of sides: '"+sides[1]+"' is not 1 or
greater\n")
     sys.exit(1)
if(float(sides[0])/sides[1] <= 0 ):
     sys.stderr.write(prog + ": sides: denominator is greater than the number of
sides '"+sys.argv[1]+"'\n")
     sys.exit(1)

ang = math.pi*sides[1]/sides[0]
cos_a = math.cos(ang)
R = 1/(2*math.sin(ang))

#ant_ht    = R*math.sqrt((1-cos_a)*2)
#pyr_ht =  R*cos_a/math.sqrt((1-cos_a)/2)
pyr_ht =  R*math.sqrt((1+cos_a)*2)/math.tan(ang)
recip_rad = R*math.sqrt((cos_a+1)/2)

#sys.stderr.write("pyr_ht=%.17f recip_rad=%.17f\n"%(pyr_ht, recip_rad))

files = " tmp1.off"
os.system("rm" + files)
os.system("polygon -t pyr %d/%d -l %.17f -T > tmp1.off" % (sides[0], sides[1],
pyr_ht))
os.system("pol_recip -a -r %.17f tmp1.off" % (recip_rad))

Hi Adrian,

I noticed you pushed some code so I tried it. Here are a few test runs.

In this model, do the diagonals connect the wrong vertices?

polygon -t ant 7 -E 1 -a 45 | antiview

In this case, 45 degrees puts it back to being a prism. So trapezohedron goes to
infinity and OFF file contains inf and -inf. This is ok, but it look like an
error to antiview

polygon -t ant 4 -T -a 45


The error message has a spelling problem

polygon: error: option -T: only a pyramid or an antiprisms can be made into a
trapezodron


It might be nice to allow only one side of the antiprism be augmented.

Roger

Hi Roger

On Wed, 7 Jul 2010, Roger Kaufman wrote:
> In this model, do the diagonals connect the wrong vertices?
>
> polygon -t ant 7 -E 1 -a 45 | antiview
>
> In this case, 45 degrees puts it back to being a prism. So trapezohedron
> goes to infinity and OFF file contains inf and -inf. This is ok, but it
> look like an error to antiview
>
> polygon -t ant 4 -T -a 45

It was difficult to decide how the -a argument would work. In order
to preserve the symmetry alignment both polygons need to move. If
it is the angle that the polygons move relative to each other then
the polygons appear to move only half of that angle (because
they both move.) I therefore chose to make it the angle that the
polygon moves relative to the symmetry of the antiprism.

That means that the prism angle for the first one 180/7, and
the second one is in prism position with 45.

I don't get the infinite vertices here. I have just pushed
a check for them.


> The error message has a spelling problem
>
> polygon: error: option -T: only a pyramid or an antiprisms can be made into a
trapezodron

I have changed this.


> It might be nice to allow only one side of the antiprism be augmented.

They are interesting as self-dual polyhedra. Do you know if they
have a name? Maybe I should include the trapezohedra constructions
by name, e.g. -t trapantiprism, -t trappyramid then I could also
include this one by name (and there are other models that could
be included in the future).

Adrian.
--
Adrian Rossiter
adrian@...
http://antiprism.com/adrian