Is this a V6 hex dome?

[Shereef Bishay]

Folks, can someone please help me identify this dome:

https://youtu.be/js3Bp_3glvY?t=65

The builder in the video claimed to have built it without a plan! Which is impressive. 

How can I go about figuring out the right measurements for this. I'm really inspired by it and want to build one!

Any help would be greatly appreciated.

[TaffGoch]

Sheeref,

That is, indeed, a 6-frequency dome -- Specifically, the DUAL of a 6v{3,3} icosahedral triangle tessellation.

The blue lines depict the triangular edges of a 6v{3,3} icosahedron tessellation.
This image is a video frame grab of the "Hawaiin" dome, built by the same guy.
He has posted a YouTube video of that dome, as well.

For more info, wikipedia has coverage on icosahedral polyhedron triangle-tessellations and their duals.

Also:

Dual Geodesic Icosahedron Pattern 6v{3,3}

-Taff

(aka, David Price)

[Shereef Bishay]

Taff, thank you so much! What a generous response. 

The link you sent is excellent. I'm having trouble using it though. It identifies 9 different edges sizes but doesn't map them onto the diagram itself. What am I missing?

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[Levente Likhanecz]

hi,
try to import the attached dae (collada) file into sketchup, wher u
can take measurements.
supposed to be 1000unit radius.
my windows rig died, i cannot check if proper looking.
it was created by antiprism (Adrian Rossiter's)
command for it was :
off2dae -x ve geo_3_3_d > geo33d.dae

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[Levente Likhanecz]

the geo_3_3_d built in model in antiprism looks like the stuff Taff linked

[Levente Likhanecz]

hi Shereef,

i attached a new pic where the edges supposed to be colored same
length same color.
and the green numbers are edge index numbers.
also attached and excel sheet, where you can search the lengths by
entering indexnumbers

hope u can figure out.

cheers, lev

[Levente Likhanecz]

hi Shereef,
screensot from Libreoffice calc srpreadsheet (ods file)
looks like the excel export miss some function.
this spreadsheet require to install a plugin to libreoffice calc to
display the colors.
(xbackground.oxt the plugin)

On 9/17/20, Shereef Bishay <she...@gmail.com> wrote:

[Blair Wolfram]

I identify this as a Class 2, 9 frequency tesselated dodecahedron. 

When triangulated as shown in the attached drawing, it's dual is a Class 2, 9 frequency geodesic breakdown.

Blair

To view this discussion on the web visit https://groups.google.com/d/msgid/geodesichelp/CAOkvEDmDOuMaNJ3Gy0a0vP821U%2BCMYUNyS3%2B%3DymZn6r6xS7YAg%40mail.gmail.com.

[Blair Wolfram]

Make that Class 1, not Class 2

Blair

[Levente Likhanecz]

hi Shereef,
the geo_3_3_d antiprism model and the "orange ball" Taff linked has
differencies.
the orange ball has less differents over the spread of strutlengths.

on the "orange ball" page has a hidden list of coordinates.
i succeeded to transform those coordinates into visuals, and now the
attached assembly diagram and the strut lengths are corresponding to
the orange ball.

also attached the translated off format file what u can display in
antiprism with the following command

off_color -e S taff33.off | antiview

cheers, lev

[Levente Likhanecz]

Shereef,
the provided vertices coordinates for the orange ball are not on
sphere. all of them slightly above radius 1, and they vary lightly.
the antiprism version must be radius 1 all vertices.

[Shereef Bishay]

Thank you so much!

Which dome architecture could provide a similar look and geometry with the least variety between edge sizes?

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[Levente Likhanecz]

Shereef,

"Which dome architecture could provide a similar look and geometry
with the least variety between edge sizes?"

i can say only the differences these 2 versions.
thats antiprism (10 sizes)
0.100910507606437
0.105540906087706
0.114445458450013
0.125405678576265
0.135126518203216
0.136199281557703
0.144190573127377
0.147266064140502
0.151510555209234
0.158028871815886

thats Taff's link orange ball (9 sizes, 1 less on similar geometry)
0.110479812634272
0.119570269559425
0.130980127353853
0.133299528223126
0.136105099162739
0.139272991195234
0.139635312288118
0.139827912938049
0.140745155773748

so the orange ball is the winner.

[Adrian Rossiter]

Hi Lev

On Sun, 20 Sep 2020, Levente Likhanecz wrote:
> "Which dome architecture could provide a similar look and geometry
> with the least variety between edge sizes?"
>
> i can say only the differences these 2 versions.
> thats antiprism (10 sizes)
> 0.100910507606437

...

> thats Taff's link orange ball (9 sizes, 1 less on similar geometry)
> 0.110479812634272
> 0.119570269559425
> 0.130980127353853
> 0.133299528223126
> 0.136105099162739
> 0.139272991195234
> 0.139635312288118
> 0.139827912938049
> 0.140745155773748
>
> so the orange ball is the winner.

geo_3_3 has its vertices on a sphere, geo_3_3_d has its faces
tangent to a sphere, the model Taff posted looks like the canonical
version, with its edges tangent to a sphere

canonical geo_3_3_d | off_report -C E

[edge_lengths_cnts]
0.11047981227853862 = 60 (range +/- 2.959943713742863e-09)
0.11957026939225374 = 60 (range +/- 2.4150332914030237e-09)
0.13098012705223094 = 120 (range +/- 2.6695973348589419e-09)
0.13329952804102058 = 60 (range +/- 2.7889498349642494e-09)
0.13610509933362266 = 120 (range +/- 2.6485896531536213e-09)
0.13927299106538826 = 120 (range +/- 2.6899712596284431e-09)
0.13963531261865786 = 30 (range +/- 2.5982424961545902e-09)
0.13982791299152059 = 120 (range +/- 2.7287266196607263e-09)
0.14074515588753145 = 120 (range +/- 2.9005534224468832e-09)


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

[Levente Likhanecz]

oh Adrian, my bad,
i'm very sorry that i offended your work. especially the offend based
my lack of knowledge/stupidity.

so is it impossible, to have all vertices exactly on sphere while
maintaining each hexagons "planar/flat"?

lev

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[Chris Kitrick]

Lev, 

It is possible to have all the hexagon boundary points on the sphere such that all the hexagons are planar (flat).

Attached is the diagram of a solution. No geometry data is included. Each half triangle is right spherical triangle. The solution for class I is quite straightforward.

There are nine different regular triangle edge lengths: four unique hexagon boundary lengths, five different radial lengths.

There are five unique hexagons. Each has a different radial size.

Chris

[Chris Kitrick]

[Adrian Rossiter]

Hi Lev

On Sun, 20 Sep 2020, Levente Likhanecz wrote:
> oh Adrian, my bad,
> i'm very sorry that i offended your work. especially the offend based
> my lack of knowledge/stupidity.

Your message was fine, so please don't apologise! I just thought
you would be interested in the properties of the different models.

[Levente Likhanecz]

thank you Adrian,
after your message i dig into this canonical stuff over the net and as
well your canonical man pages.
your antiprism package is really-really dense. solutions for
everything, and endless tricks with the paramers/options/switches.

i'm little impatient when i see some new face seeking solution and i
jump in the middle, and i know many people here have lot of work to do
outside this forum, them busy with other aspects of living.

fortunately a little later the "big guns" appear and making things
strait, for everybodies' satisfaction.
lot to learn, lot to adopt.

cheers, lev

On 9/20/20, Adrian Rossiter <adr...@antiprism.com> wrote:

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[Levente Likhanecz]

thanks Chris,

is there geometrical solving (compass/ruler-sketchup) or iterativ mayhem only?
(gosh i have to assemble a new windows rig i really suck with linux
applications that mimics excel or anything else from windows)

cheers, lev

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[Levente Likhanecz]

Hi Adrian,
i stumbled again into your da vinchi bridge style rotegrities (Ashok
brought up a flatband/ or prism cross-section geo_2)

so i pulled up a geo_2_2_d
rotegrity -a n,0.35 -c s -O n geo_2_2_d | antiview -v 0.005 -e 0.005
but i cannot interpret the resulted length fractions and angles (for
drill holes)

this i suppose the strut pipe radius for accurate touching
radius: 0.004716357441961 once i replace to antivew
-e(dge) it will exact touch

the next is the shortest strut where still reach all struts
strut length from: 0.302685530194476

then pieces of each lengths also clear

but the last 2 lines fractions and angles is twilight zone for me.
i cannot locate the old thread where you introduced this feature.


cheers, lev

On 9/20/20, Adrian Rossiter <adr...@antiprism.com> wrote:

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[Adrian Rossiter]

Hi Lev

On Mon, 21 Sep 2020, Levente Likhanecz wrote:
> rotegrity -a n,0.35 -c s -O n geo_2_2_d | antiview -v 0.005 -e 0.005
> but i cannot interpret the resulted length fractions and angles (for
> drill holes)

...

> but the last 2 lines fractions and angles is twilight zone for me.
> i cannot locate the old thread where you introduced this feature.

Look at unit type 0 like this

rotegrity -a N,0.35 -c u -O N geo_2_2_d -m map_0=red | antiview -v 0.005 -e 0.005

Result
------
Radius and range of radius valaues (model units), followed by minimum
strut length to ensure contact, then three lines per strut type
1: strut_index_number (used for colouring), number_of_struts
2: strut_length (model units), fraction_1st_contact, fraction_2nd_contact
3: ang_1st_contact, ang_2nd_contact, ang_3rd_contact, ang_4th_contact
In the coloured model, angles are measured anticlockwise from the white
end

radius: 0.004716357441961 (+/-0.000000000000000)
strut length from: 0.302685530194476
0 , 60
0.3500000000000, 0.0675920997222, 0.3558640332407
-171.91511979, -10.51369349, 10.51609248, 171.91511979
...


Length measurements are along a line segment between end-ball centres,
with contact points projected radially (with respect to the unit's
cylinder) onto this line. Angle measurements are are around the
central line, from a 0 pointing "outwards" (consider the angles to
be relative to each other).

In the second line for strut type 0

0.35: strut length
0.35 * 0.0675920997222: length from end to first contact
0.35 * 0.3558640332407: length from end to second contact

Look along the strut from the white end, you can draw a
radial line to each contact point, these angles, in order
that each contact point is encountered, are
171.9 clockwise, 10.5 clockwise, 10.5 anticlockwise, 171.9 anticlockwise.

I have attached an image.

I notice that in some nexorade reports all the strut types are
labelled as 0, rather than being numbered sequentially. I have
pushed a fix for this.

Also, the results have not been verified! You might want to
check they are approximately correct in Sketchup if you are
considering making a model based on them.

[Levente Likhanecz]

uh, thanx Adrian,
so, with the sight of b/w endballs the angles are clear (more or less).

but the length fractions are still foggy. it looks to me that the
lengths in this command sequence are all strut fixed to (0.35) length,
choosen by me.
are these struts symmetrically placed (centered between frac points 2
and 3) and on your attached pic i measure frac distances from both
ends?
from white ball end frac 1 and 2. then
from black ball end frac 4 and 3. ????

(sorry taking your time instead of setting up my new rig with
sketchup, my linux runs on 1st gen atom with limited hardware
capacities to force wine/sketchup on it)
but soon i have to, its not easy once i got addicted to windows.

cheers, lev

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[Adrian Rossiter]

Hi Lev

On Tue, 22 Sep 2020, Levente Likhanecz wrote:
> but the length fractions are still foggy. it looks to me that the
> lengths in this command sequence are all strut fixed to (0.35) length,
> choosen by me.
> are these struts symmetrically placed (centered between frac points 2
> and 3) and on your attached pic i measure frac distances from both
> ends?
> from white ball end frac 1 and 2. then
> from black ball end frac 4 and 3. ????

if 0 is the white ball centre label, and 5 the black ball centre label then

length 0->1 = length 4->5 = 0.35 * 0.0675920997222
length 0->2 = length 3->5 = 0.35 * 0.3558640332407

Changing the struct length changes the fractions, although the contact
points themselves are fixed (the model is effectively the same, but with
the strut ends extended or shortened).

[Levente Likhanecz]

thanx, totally clear.

cheers, lev


On 9/22/20, Adrian Rossiter <adr...@antiprism.com> wrote:

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[Adrian Rossiter]

Hi Lev

On Tue, 22 Sep 2020, Levente Likhanecz wrote:

> thanx, totally clear.
...

>> On Tue, 22 Sep 2020, Levente Likhanecz wrote:
>>> but the length fractions are still foggy. it looks to me that the
>>> lengths in this command sequence are all strut fixed to (0.35) length,
>>> choosen by me.

It may be that it would be better to give lengths for nexorade struts
in model units rather than fractions of the struct length, as the strut
radius is given in model units.

I'll review the report based on your feedback and see if I can
improve it and/or provide a clearer explanation.

[Chris Kitrick]

Lev,

There is only a simple iteration to solve the flat configuration of this class I hex/pent configuration.

Below are all the spherical angles (in degrees) for all the labeled right spherical triangles. From these you 

can generate chord factors, etc. The image is also attached for reference. 

Chris

Right

Spherical

Triangle a b c A B C

----------------------------------------------------------------------------------------------------------

t[00]       3.553711440822  2.578525175352  4.389661472179 54.080004313636 35.999999999999 90.000000000000

t[01]       6.834457847956  2.578525175352  7.302536379205 69.425611380982 20.728385258139 90.000000000000

t[02]       6.084041403983  4.046366665742  7.302536379205 56.494384305383 33.720741822183 90.000000000000

t[03]       6.391704213641  3.539091538968  7.302536379205 61.142976945229 29.054694637655 90.000000000000

t[04]       6.723588499164  3.539091538968  7.594356471029 62.362638749389 27.845319767203 90.000000000000

t[05]       6.514490666570  3.911735747030  7.594356471029 59.145366825843 31.077340116398 90.000000000000

t[06]       6.933642498040  4.046366665742  8.022995142563 59.875534090283 30.369702011965 90.000000000000

t[07]       7.010223195830  3.911735747030  8.022995142563 60.979099083874 29.260595976070 90.000000000000

t[08]       7.037872451809  4.046366665742  8.113092799663 60.248931819435 30.000000000000 90.000000000000

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[Adrian Rossiter]

Hi Lev

On Tue, 22 Sep 2020, Adrian Rossiter wrote:
> It may be that it would be better to give lengths for nexorade struts
> in model units rather than fractions of the struct length, as the strut
> radius is given in model units.
>
> I'll review the report based on your feedback and see if I can
> improve it and/or provide a clearer explanation.

I have amended the report to look like the following


rotegrity -c s -a N,0.35 -O N geo_2_2_d | antiview -v 0.005 -e 0.005

Result
------
Radius and range of radius valaues, followed by minimum strut length

to ensure contact, then three lines per strut type

1: strut_index_number (used for colouring), number_of_struts, strut_length
2: length_to_1st_contact, length_to_2nd_contact,
length_to_3rd_contact, length_to_4th_contact
3: angle_of_1st_contact, angle_of_2nd_contact,
angle_of_3rd_contact, angle_of_4th_contact
Lengths are measured from either end of the strut
Angles (degrees) are measured looking along the strut central line
from the white end (display with -c u), anticlockwise, from an outward
pointing zero angle (consider the angles to be relative to each other)

radius: 0.004716357441961 (+/-0.000000000000000)
strut length from: 0.302685530194476

0 , 60, 0.3500000000000
0.0236572349028, 0.1245524116343, 0.2254475883657, 0.3263427650972
-171.91511979, -10.51369349, 10.51609248, 171.91511979
1 , 60, 0.3500000000000
0.0264519444420, 0.1254839814807, 0.2245160185193, 0.3235480555580
-172.77070433, -11.02595440, 11.02830909, 172.77070433
2 , 60, 0.3500000000000
0.0271348655444, 0.1257116218481, 0.2242883781519, 0.3228651344556
-171.18102165, -9.35237819, 10.02782188, 171.18102165
3 , 60, 0.3500000000000
0.0271636854163, 0.1257212284721, 0.2242787715279, 0.3228363145837
-173.79644006, -11.96659964, 12.63956827, 173.79644006
4 , 60, 0.3500000000000
0.0312087916890, 0.1270695972297, 0.2229304027703, 0.3187912083110
-170.64258383, -8.97032796, 13.84683775, 170.64258383
5 , 60, 0.3500000000000
0.0508139070263, 0.1336046356754, 0.2163953643246, 0.2991860929737
-172.34320112, -12.38526070, 12.38526070, 172.34320112

[Levente Likhanecz]

hi Adrian,
yep the strut radius in model unit nicely works for me, i could
quickly figured it out how to calculate from it buildpipe/sphere
radius (i hope my theory is ok)

let say i choose a standard 1 inch pipe (33,7mm OD)
OD / (2 * strutR) = sphere radius
then
sphere radius * (0.35)= actual strut length
then come the fractions as you described.

i guess this is the way (strut dia and sphere radius must be rationed)


and the node fraction part after the explanation is clear, may be
easier just to update the man page and the report's verbal explanation
of the result lines.

or if you consider change to unit lengths instead of fraction, for me
would give a more consistent feeling to print all 4 lengths from 1 end
only - to look similar to the 4 angles.

example:
verbal description
strut radius: xxxxxx
minimum strut length:xxxxxx
fixed(actual) strut length: (0.35) <- this repeated presently with all struts

0, 60
length1 length2 length3 length4
angle1 angle2 angle3 angle4

next strut .......

thanx, cheers, lev

On 9/22/20, Adrian Rossiter <adr...@antiprism.com> wrote:

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[Levente Likhanecz]

thanx Chris,

i will study it throughoutly.
i hope Shereef also follows up the thread he popped, because this
seems to be the feasible easiest solution to his question.

best regards, lev

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[Levente Likhanecz]

hi,

so i crunched Chris'es hexadome into a sketchup model.
the radius is 10,000 with the 6 additional decimals, so measurable to 10 digit.
penta/hexa tips on full radius and the set-aside components are all
flat. (as sketchup considered them as flat, it must be flat)

happy building, cheers lev

[Levente Likhanecz]

hi Shereef,

here is the "higher precision" sketchup model.
14 digits measurable (8+6) in sketchup.
please note, i left the hexagons and penta mid-vertices on full radius.
if you redraw the circumferences it will create you flat backsides.

cheers, lev

On 9/17/20, Shereef Bishay <she...@gmail.com> wrote:
> Folks, can someone please help me identify this dome:
>
> https://youtu.be/js3Bp_3glvY?t=65
>
> The builder in the video claimed to have built it without a plan! Which is
> impressive.
>
> How can I go about figuring out the right measurements for this. I'm really
>
> inspired by it and want to build one!
>
>
> Any help would be greatly appreciated.
>

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[Blair Wolfram]

Hello Taffgoch David White,

 

You are absolutely the best at what you do!

I sense there are two definitions of ‘frequency’ being used in discussion.

 

Background:

The platonic solid icosahedron is a 20 sided polygon with each vertex or hub on the surface of a sphere. Each face is an equilateral triangle, referred to as a  parent triangle. In geodesics this is sometimes (maybe incorrectly) identified as a one-frequency 1v sphere, there is one frame member connecting any two pentagons center to center, and all members are the same length. There are three rows of triangles in this sphere, using a formula frequency times three as the denominator of the spherical fraction, a complete one frequency sphere is a 3/3^rd sphere. If one row of triangles is removed so the sphere sits flat it becomes a dome,  a 2/3 sphere high profile dome. All frame members or struts are the same length, and if this length is 8’, a one frequency 8’ strut will build an 11’ diameter dome.

 

If the one strut connecting two pentagons is divided into two struts, it becomes a 2 frequency 2v dome. Dividing the edge of the parent triangle in two requires an additional strut length to raise the new vertex to the surface of the sphere. Two struts separating pentagon centers and a maximum strut length of 8’ will build a 26’ diameter sphere with 2 strut lengths. 2v x 3 = 6; a full sphere is a 6/6 profile. A half sphere is a 3/6 dome and can build a 26’ diameter dome.

 

The first three frequency domes were widely publicized in the 1960’s in Mother Earth News, Popular Science and other publications and made with Fuller’s 3 strut 3v design, as is Bucky’s personal dome in Carbondale, IL. This sphere division doesn’t have a horizontal truncation line, so the 3 strut 3v dome requires 15 triangles removed around the base. Fuller took the 60 remaining triangles in a low profile dome as a fraction of the number of triangles in the whole sphere 180 and identified this using the fraction denominator 8, low profile 3/8 high profile 5/8; 3 strut 3 frequency dome. This is the Class One Method One geodesic used by Monterey, Cathedralite, Timberline, Pease and early dome manufacturing pioneers. The Class One Method One 4v dome has 5 struts and sits flat at the half sphere only.

 

David Kruschke added a 4^th strut to the 3v dome so the dome sits flat without removing any triangles, and has 9 complete rows of triangles. 3v x 3 = 9. These domes are identified as 4/9 low profile or 5/9 high profile 3v 4 strut domes, and are Class One Method Two geodesic. Both Fullers 3 strut and Kruschke’s 4 strut divisions are three frequency domes. The Class One Method Two 4v dome uses 6 strut lengths and sits flat at 5/12, 6/12 and 7/12 truncation. Both Bucky’s 5 strut and Kruschle’s 6 strut are 4 frequency domes.

 

Question:

When you illustrated the Orange Dome, (modified version attached) you identified it as a 6 frequency dome, when I understand it to be (the dual of) a 9 frequency dome.

Are you using the number of unique strut lengths to identify this as a 6 frequency dome, and not identifying frequency as the parent triangle edge divisions?

 

Blair

 

 

 

Blair F. Wolfram

thedo...@domeincorporated.com

612.333.3663

www.domeincorporated.com

On Thu, Sep 17, 2020 at 9:34 PM TaffGoch <taff...@gmail.com> wrote:

Sheeref,

That is, indeed, a 6-frequency dome -- Specifically, the DUAL of a 6v{3,3} icosahedral triangle tessellation.

The blue lines depict the triangular edges of a 6v{3,3} icosahedron tessellation.
This image is a video frame grab of the "Hawaiin" dome, built by the same guy.
He has posted a YouTube video of that dome, as well.

For more info, wikipedia has coverage on icosahedral polyhedron triangle-tessellations and their duals.

Also:

Dual Geodesic Icosahedron Pattern 6v{3,3}

-Taff

(aka, David Price)

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[Dick Fischbeck]

F=root ((n-2)/10)

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[Dick Fischbeck]

2003 archives:

Euler says:

v+f=e+2

By knowing v, the number of vertexes(your cells), we know
the ratio of faces and edges. For omnitriangulated
polyhedra, there are 3 edges per 2 faces per each (vertex
minus 2).

10f^2+2=n,

therefore,

f=root[(n-2)/10]

Each cell can have the same alpha, angular deficit.

720/n=alpha

[Blair Wolfram]

Dick;

 3v sphere has:

v = vertices = 92

e = edges = 270

f = faces = 180

v+f=e+2 

92 + 180 = 270 + 2

272 = 272

Okay so far.

Take me through the next formula; what is n, and does root mean square root?

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[Adrian Rossiter]

Hi Blair

On Tue, 29 Sep 2020, Blair Wolfram wrote:
> I sense there are two definitions of *‘frequency’* being used in
> discussion.
...

> When you illustrated the Orange Dome, (modified version attached) you
> identified it as a 6 frequency dome, when I understand it to be (the dual
> of) a 9 frequency dome.

The operation you have applied to determine frequency is called
"kis" in Conway notation (join face centers to edges to make
triangles), rather than being a dual operation (convert faces
to vertices, and vertices to faces).

https://en.wikipedia.org/wiki/Conway_polyhedron_notation#/media/File:Conway_relational_chart.png

The kis operation increases the number of edges by a factor of three,
while the dual operation does not change the number of edges, and so
reading the frequency from the kis model gives a higher value than
reading it from the dual model.

[Levente Likhanecz]

oops, Adrian, this message i missed somehow.
it looks clearer (to me). maybe to other people as well.

the drill-angles i could not research yet, as my virtualized sketchup
cannot handle the full collada (struts as cylinders and vertices as
balls - way too much polygons to move). the openGL driver is not
working too well with intel atom linux.
for this i have to dig up my old desktop, with discreet nvidia graphics.

thanks and cheers, lev

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[Adrian Rossiter]

Hi Lev

On Wed, 30 Sep 2020, Levente Likhanecz wrote:
> oops, Adrian, this message i missed somehow.
> it looks clearer (to me). maybe to other people as well.

Thanks for letting me know.