Giza
 Octagon

 ```wondering how octagons fit just so: i'll make a template octagon object as i've already done with a hexagon and i could start from scratch, but i want to maintain some continuity so, i'll just import that object and make an octagon out of that ```
 ```overhead, rendered wireframe composite of the hexagonal object group i made... full, selected red outline (below right) ...and a tight zoom of the far left corner (below, far left) showing individual objects ```
 ```again... here's my hexagon worksheet which is fun to just stare at... ```
 ```so, in making an octagon from my hexagon: using the same objects i have, already... using the same textures, so they'll match (and i can always change the color later) ```
 ```my hexagon object group measures 41 x 35.64 units ...discounting the thickess of the sides and corners... which are each 1 unit thick (twice one half, each side) point to point, it measures 40 x 34.64 = 2 (20 x 17.32) ```
 ```i have it further grouped by sides and corners to change those categorically, anytime (above) but i'll just ungroup this whole thing (below) ...keeping only the top side and its 2 corners ...deleting the rest (they'll only confuse me) ```

 ```so... just keeping the 2 corners and one side from the hex object... ```
 ```group those together, duplicate... and put the new one negative Z from center i now have 2 identical sides (as groups) opposite center, positive and negative Z each with 2 rounded corners (cylinders)... and when grouped, together that's centered ```
 ```what remains, grouped together or not ...their mutual Z coordinate is 17.32 ``` ```...that is: up, or north, from center (as: the top side of a hex centered) ```
 ```now, duplicate that group and rotate it 90° group both of those, duplicate n rotate 45° so, now i have eight sides, all centered... ``` ```and with twice as many corners as i need... i'll delete the extras later, when i'm done ```
 ```here's my octagon sides... but too long... first, ungroup all of this... just once... not to first level, of 8 individual groups ...or zero level, of individual objects... ...i want 4 pairs of sides, at 90° and 45° ``` ```select just one group and reduce its width (below) until dots match top line of 17.32 ```
 ```the side's width remains one unit only its length is being adjusted ``` ```...and it's still proportional... with the dots being squashed here what i'm aiming for is the center ```
 ```and then there's this: little freaky thing... i almost didn't notice ``` ```...see, it started at 21 ...and i just reduced it a couple units at a time ``` ```...it landed right on it ...at an exact even unit so that got my attention ``` ```cuz i started with a hex 20 proportioned >> 17.32 ``` ```the line groups measure: ...15 units wide x 35.64 ``` ```and the dot's new coords ...close: (-7.20, 17.30) but my target is: 17.32 ```
 ```so that's interesting ...stumbled upon some ...basic, fundamental ...geometry, there... ``` ```proportional to the hex sides of: 20 x 1 but, with an additional 1 unit (2 halfs) (for the radii, of the cylinder corners) the actual side, point to point is 14.29 ```
 ```...and... just noticing: it's close enough at an even 15... ``` ```which, in half, is 7.5 over 10 is 0.75 or 3/4 ``` ```but that's the whole group... see, the program is computing the proportions automatically ``` ```...notice the polygon's sides are still true... 16 sided, right edge on ```
 ```...the dot is squashed to 0.71 which, i can only assume to be rounded up from .707 sine wave or anyways, close enough again ```
 ```so, back to the group, up from 15 stretching the group out to 15.07 puts the dot right on the line... so, that would probably be 0.0707 ```
 ```ok, so now i select all 4 of these groups and make them all that same 15.07 wide... ...they were, already, all still 35.64... ``` ```(17.32 times 2, plus 1 for 2 halfs thick) ```
 ```this is where i ungroup everything down to individual objects and return them all to their desired 1 unit width (diameters) ```
 ```the circle and square are each 20 side to side (to be even with the hex side, point to point) ```
 ```so now i have an octagon ...made out of a hexagon with the same proportions ```
 ```so now, i can just group them together and adjust them to any common diameter ...of any relevant arc, for comparison ```
 ```done the normal way... same process as before ``` ```just the corner dots this time: grouped, duplicated and rotated ```
 ```ungroup all those to zero level: then regroup and rotate to 22.5° ```
 ```no matter what i rotate them to ...their centers will still be right on the circle edge line ```
 ```but i want pairs with common coords so i can put a cube line object in: ```
 ```regrouped, the whole area is 19.48 square that's how far the one unit diameter dots approach the 20 side circle-square's edge (at their angular displacement of 1/16th) ```
 ```the actual points, their centers 9.24 units... relative to center and that's where i put the sides ``` ```length of the line between them would be double that coordinate ...which... works out to: 18.48 ...which is one less than 19.48 (the 1 being 2 halfs diameters) ``` ```...the other coordinate is 3.83 ...the octagon's sides' lengths being... double that... at 7.66 ```
 ```that 3.83 x 8 = 30.64 30.64 - 18.48 = 12.16 and 12.16 / 4 = 3.04 for a 20 sided square ``` ```for a 10 sided square ...that would be 1.52 ``` ``` for a 1 sided square that would be 0.152 ```
 ``` 1/8 = 0.125 3/16 = 0.1875 5/32 = 0.15625 = 10/64 = 20/128 = 40/256 = 80/512 ```
 ``` 0.15625 - 0.152 = 0.00425 x 4 = 0.017 ```
 ``` 1/64 = 0.015625 9/64 = 0.140625 ``` ``` 1/128 = 0.0078125 19/128 = 0.1484375 ```
 ```anyways, i put my sides in at 9.24 ...and, each with a length of 7.66 repeat the aforementioned process: ...rename, regroup, retexturize... ...and now, compare to the circle: ``` ```but, what i have is not a common octagon that we're used to seeing in a stop sign (with sides flush to: up, down and side) ...this one is 20 units, from the points so, the corners touch the circle's edges ```
 ```...side to side it's only 19.48 ...and minus 1 for 2 halfs each for the thickness of the objects makes it that same 18.48 distance between the centers of the sides ```
 ```all the dots in this image (above) are centered on the circle's edge, and all the lines are between them ``` ```now, if i wanted to do a regular stop-sign octagon... i could start over and measure from the 10 unit sides or i could just resize a duplicate of this to 21 plus ```
 ```21 puts a side at 9,96 resize again to: 20.08 ...rechecking the math ...the dots line up... right exactly at 10.00 ```
 ```and their thickness has been projected relatively to a slightly exploded 1.08 ```
 ```...reset those to an even 1.0 ...repeating the process with a duplicated corners group... ```
 ```...and here we have the regular stop sign we're used to seeing: ...a large, or outer octagon... ```
 ```these sides are centered on the edges of the circle and the square ```
 ```the half transparent pink and putple lines are where you might expect (no surprises), (1/16 of a circle relatively, or to 22.5°) but the more solid maroon and violet lines are rotated half, inbetween that normality to an additional 11.25 degrees (11.25 and 33.75 respectively) ...where their rounded corners just touch, or: are tangent to the inner sides of the outer octagon ```
 ```this is the kind of thing that's shown up occasionally before in the geometry of giza ``` ```...lines not intersecting exactly but when made a certain thickness... they present as tangent to something ```
 ```...now, again... these are all 0.2 if i made them 0.3 they'd be tangent to each others' centers ...agd again, that's... outer octagons 20 sides, inner octagons 20 corners ```
 ```so, an octagon, at that 3/4 turn, is demonstrating the 3 internally ...locking on to it, at the curve ``` ```the 3:4 ratio is natural for an octagon the 4 in this case being the sum of the 2 objects, corner and side, 0.2 each... ```
 ```...very celtic, very spirograph: ...even wth the different colors it's hard to tell where one line ends, and another keeps going... ``` ```almost as if that's what it is... it's just naturally demonstrating an inherently incremental spiral: like the golden mean or a fractal ``` ```and where it's not "exactly" exact ...is because it's not a circle... it's a square'd off version of one ...or a square's... version of one ```
 ```or it could also be said to be a circle's version of a square ``` ```the octagon is as a landmark or signpost ...inbetween... a square and a circle... ```
 ```in this one (above), with just 2 outer octagons ...the solid blue and green octos are normal... and in the one (below), adding 2 more octagons: the paler blue and green being: 11.25 and 33.75 ```
 ```and we could do this all day (until we ran out of colors) ...a natural infinity mirror ``` ```proportionally accurate only the scale projects (inward and/or outward) ```
 ```with their relative value to each other being their common phase of interest... and their lateral counterparts, sharing similarities... reflectively relatively ...this thing measures stuff for you... ```
 ```handy thing to have around ...save that object group: move on to the next thingy ```

 Giza
 Octagon