Astra - Giza

Hex 4

virtually assembling
a Crossed Double V
...or... Quater Foil...

first image, right...
start with 4 circles
(or dots, or discs)
of various colors

each cylinder object
is 20 units diameter
with varying heights
...or thicknesses of...
0.8 yellow and green
0.9 red and blue, and
1.0 white in the center

low camera angle, above: now adding a hex group (rounded version, purple)
...which i've pre-made, and saved as a user object (importable into anything)

which is still the original stock dimensions...
41 x 35.64 ...as shown in wireframe, below

so 40 x 34.64
point to point

and in Open GL mode, above
showing some transparency

the cylinders and spheres (as for the tube sides and ball corners)
...are all 1 single unit diameters... side cylinders are 20 units tall...

...now, enter another hex frame
(flat version, orange and yellow)
cylinder corners and cube sides

...and the dimensions of
the sides for this object
are the same as in the
rounded version, above

...but here, i've altered the
corner dots just a little bit...
doubling diameter to 2 units
and the height is now 1.2 tall
...leaving the sides at 1.0 tall

left image, wireframe mode
...overhead of everything...
duplicate another center circle
just for this squared hex frame
as a scaling guide (red, selected)

and this, i make larger than
everything else that's to be
group resized... collectively

...below image... wireframe
camera angle of everything

...and i'm... not really sure...
what i''ll be resizing them to
so i pick 80 units diameter...
...as a nice, round number...
though maybe 120 is better

...now, i started out with the circles as 20 units diameter...
...and the hex comes with... or starts out as... twice that...
at 40 units corner to corner (point to point, corner centers)

ah, so this will be easy... i just resize the group to 40
...and everything divides itself automatically in half...
proportions... dimensions... transported to half-land

except for
the height
or, overall
thickness

...before rendering this
...i resized the thickness
of the orange lines to 1.1
so, everybody's different
thus, no texture overlap

and this isn't exactly what i was going for
...what i need is a triangle in the middle...
...but, this is... what i have to start with...

so, it's kinda back to the drawing board
here, i assemble a triangle from the hex

...start off by
...just deleting
every other side

side lengths in the
left image are 20
below image 40

and where they want to be, to meet
just happens to be a nice round 30
compared to the other 20 and 40

right where the cylinder sides
...for the other rounded hex...
have their centers... naturally

so, that should be easy
...for these new corners
i'll just copy from those

above image, the duplicates of those...
now upright, with heights still set to 20

...left, max zoom overhead wireframe
...resizing the corners' heights to 1.4
...and while i'm at it, the sides to 1.3

and grouping together with
another hidden resize disc

that's the thingy i want
get rid of the other guys
duplicate the blue triangle
and rotate 180 degrees...

...oops... i forgot to resize them...
but look where the corner dots fall
right on the edges of the 4 circles
...20 30 40... basic geometry...

below image: implementing
the famous eyeball method

from 40, it's 23.09 for the whole group...
...downsizing to match a single 20 circle...
but, the corners are made from hex sides
each being 17.32 from scene center point
(see wireframe of triangle corner 3 above)
so that's the radius of their common circle
which i want to be 10 (or 20 diameter)...
corner dot centers matching circle's edge

to resize this accurately
i had to alter or adjust
the guide disc to 120

...which... the rounder number, there...
isn't really doing anything more for me

but the larger you start with
...the better your accuracy

so, if i want 17.32 to be 10...
for a hex, the span at 30/60�
involves the square root of 3

scaling 40 to 120
is a factor of 3...
23.09 x 3 = 69.27

...so, from 120 total, reducing diameter to 69.25
(3/4 from 70), a dot is still landing right on 10.00
not knowing what mathematical formula to go by

just going by
trial and error
one at a time

and you don't need to see pictures
but it doesn't get to -9.99 or -10.01
until... 69.24 and 69.32 respectively

...so, i was right on the edge, there
of the wiggle room window, already

and averaging those gets me 69.28
that firgure being reduced from 120

...and... 69.28 / 120 = 0.577333...
...which is the square root of...
0.33331377777777777777...
because it's a triangle, apparently

now that i've got the
lateral positioning in
(accurate to 100th)
i'll work from that

put those others
off to the side...

i'll be keeping just these 2 red dots on the lower right
...from center, 8.66 and 18.66 east (both at -5 south)

makes me think i should expand
this whole thing by a factor of 3

8.66 x 3 = 25.98 or 26
18.66 x 3 = 55.98 or 56
...if i'm de-rounding... or
re-simplifying accurately

2 x 13 meets 7 x 8
both add up to 15

...which is 3 x 5
which is exactly
what i'm doing
to the NS coord

56 x 2 = 112 for the line
total length point to point

...new line heights = 0.12
...new dot heights = 0.14

staggering at even dimensions
to avoid any texture overlap
just position the odd group
up 0.01 unit in elevation (y)

above image, the circles group is now 120

also, at a certain point somewhere in there
...i made the ground discs 1/10 thickness...
now 0.08, 0.09 and 0.10 units tall (center)

left image, 4 dots now (red, selected)

in grouping and duplicating the dots
i found something kinda interesting
their collective center is 41 (4+1=5)

and their square is 30
(plus radii makes 31)

...left image...
duplicating lines
copy and paste
...tic tac toe...
30� 60� and 90�

on the line thickness, a minor correction
i have to make them thicker than i thought
because i'm using semi-transparent textures
only parallel lines may have identical heights
and there's 6 pairs of those... so, readjusting
0.12 thru 0.17 ...and the dots moved to 0.18

the internal hexagons
measure 30 to sides...

but first, the easy way
...the old flat edge hex
measures 30 to points
...sides are ~26 apart
(top line is at 12.99)

each of those
old yellow dots
marks the center
of their new lines

...and each of the other
line crossing intersections
are marking some particular
geometrically relevant point

...below, duplicated
...stretch it out until
it's 30 wide to sides

...cyan corner dots and...
thinner bright white sides

...from 80, originally
and then down to 60
now, back up to 69.3
puts the lines on 15.0

and the points, to
right about where
...they should be...
at... 17.32 exactly

Giza

Hex 4

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