Unknown properties of the Golden Spirals disclosed:
From the true golden spiral's ellipses, to the spiraloid, up to
the anti-spiral and all the way down to the spinning black holes.

Although clearly proven to be an er­ror in math­e­mat­ics and ge­om­e­try, not a few peo­ple re­main con­vinced that the gold­en spi­ral can be de­fined by square arcs in­side rec­tan­gles which, ex­pand­ing by the ra­tio of φ, ro­tate a­round them­selves.
A ge­o­met­ri­cal­ly fas­ci­nat­ing se­quence, but which can only con­cern the squares in their con­fig­u­ra­tion of new rec­tan­gles in scale; ap­ply­ing cir­cle arcs to it and then de­fin­ing it as a spi­ral is an a­buse, al­though prob­a­bly born of an ir­re­sist­i­ble temp­ta­tion. In­deed, it would seem that the very be­ing at­tracted by this se­quence of pol­y­gons in self-gen­er­at­ing am­pli­fied ro­ta­tion was the first step to in­spire the idea of the gold­en spi­ral, with such en­thu­si­asm as to then want to ap­ply it to the tri­an­gle as well.
In re­al­i­ty, the fun­da­men­tal con­di­tion and char­ac­ter­is­tic of the spi­ral is that eve­ry point a­long its path can nev­er be e­qui­dis­tant from the cen­ter like any oth­er point; and this is true for both the rec­tan­gle and the tri­an­gle.
In ad­di­tion to am­big­u­ous scho­las­tic in­for­ma­tion, this in­cor­rect prac­tice has fu­eled bad in­for­ma­tion on the web also be­cause in fact, ex­cept for a few math­e­ma­ti­cians with their own and dif­fer­ent for­mu­la­tions, the tools to com­mon­ly and eas­i­ly trace this kind of mag­nif­i­cent curve have been lack­ing; some­times even to un­der­stand its true mean­ing if not with the help of ref­er­ences to con­for­ma­tions of na­ture.

I too found my­self em­bar­rassed by the more re­cent more de­mand­ing ap­proach, I could say start­ing from scratch, hav­ing had to tou­ch on the sub­ject in writ­ing my es­say on the ef­fec­tive def­i­ni­tion of π ("«2x2=3,14»"), which o­rig­i­nat­ed from my first dis­cov­ery in 2003 of the φ scaled cir­cles con­figuring the ma­jes­tic Gold­en Tri­an­gle from which eve­ry­thing is de­rived.
It was the in­ev­i­ta­ble stim­u­lus that took my ef­forts one step fur­ther.
Fol­low­ing this re­search, a chal­lenge that re­mained sus­pend­ed but which I could no long­er ig­nore af­ter hav­ing men­tioned it turned out to be pre­cise­ly the au­then­tic Gold­en Spi­ral, of which these cir­cles have re­mained the back­ground ter­rain.
By ad­dress­ing 'Es­sen­tial Ge­om­e­try' so­lu­tions on the web­site that in­ter­ests fur­ther and spe­cial de­vel­opments, I fi­nal­ly delved into the cor­rect, sim­plest and most de­fin­i­tive way to for­mu­late and draw said spi­ral in each mo­dal­i­ty (since there is not only one) on a Car­te­si­an sys­tem; I solved it at the root with a u­ni­ver­sal al­go­rithm well il­lus­trat­ed and offered free in its web code, more­o­ver re­veal­ing at least two 'se­crets' con­cern­ing the ge­o­met­ric as well as the cre­a­tive pow­er of the end­less­ly open fig­ure.
First of all, the pre­cise and un­prec­e­dent­ed re­la­tion­ship that binds or con­nects the spi­ral to pro­gres­sive el­lipses.
A sum­ma­ry of the re­sults is clear­ly vis­i­ble in the rich and ad­vanced PDF offered here, ex­plained in eve­ry de­tail at the ap­pro­pri­ate do­main.
Af­ter that I en­sured that from to­day an­y­one can draw and seg­ment the ex­act gold­en spi­ral, in var­i­ous ways and per­spec­tives use­ful for re­search, also with the dis­play of the cal­cu­la­tions, not only di­rect­ly from a spe­cial con­sole on the web page
but also in its own web­site (a­dapting my Ja­vaS­cript code there), with a choice of more than 5 well de­fined and il­lus­trat­ed gold­en ra­ti­os.
But that is not all.

The re­search, with the in­i­tial in­ten­tion of dis­mantling the un­found­ed claim to piv­ot a spi­ral on the gold­en tri­an­gle de­rived from the pen­ta­gon and on its gold­en sec­tionings, has led to the sen­sa­tion­al dis­cov­ery of a prin­ci­ple that we would nev­er have sus­pected ex­ist­ed, since it dem­on­strates an im­plic­it ten­den­cy of each tri­an­gu­lar ge­o­met­ric com­bi­na­tion, how­ev­er ir­reg­u­lar it may be, to a­chieve an i­de­al gold­en con­for­ma­tion, ex­act­ly that of said al­ready per­fect tri­an­gle, if only pro­gres­sive­ly am­pli­fied by spi­raloid traits ac­cord­ing to the dy­nam­ic lay­out that one im­prop­er­ly at­tempts to at­trib­ute to the so­-called 'sub­lime' one.

How the Ellipses integrate the classic Golden Section Spiral


How a triangle evolves by applying a Spiraloid

I have directly called it a 'spi­raloid' pro­ce­dure, not a spi­ral, car­ried out and il­lus­trat­ed step by step up to a suf­fi­cient pre­ci­sion lim­it on the site, which is al­ways sub­ject to up­dates.
This in­verse ap­pli­ca­tion - where it is not the spi­ral that de­vel­ops on the gold­en tri­an­gle sub­di­vi­sions, but it is the tri­an­gle that per­fects it­self un­der the pace of the spi­raloid - seems to pos­i­tive­ly re­verse the ef­fects of that ge­o­met­ric com­pro­mise whose dif­fu­sion re­mains un­der ex­plic­it crit­i­cism.

In fact, the ex­traor­di­nary rev­e­la­tion of the in­con­tro­vert­i­ble out­come leads us to fo­cus on a la­tent law that only ge­om­e­try could sub­li­mate, a har­mon­ic fac­tor ev­i­dent­ly im­plic­it in the cre­a­tive prin­ci­ple, which in part jus­ti­fi­es, if not val­i­dates, that in­stinc­tive pro­ject­ing of the hu­man be­ing, per­haps un­con­scious but not un­found­ed, to­wards com­mun­ion with the truth how­ev­er hid­den and e­lu­sive; even when he does not glimpse the ac­tu­al goal and can only run a­ground on im­pro­vised mi­rages, as it goes with this symbol.
Na­ture, for its part, al­ready knows it in it­self and ac­com­plish­es.

A further study of the spe­cial gold­en prop­er­ty of said tri­an­gle was then de­vel­oped in a care­ful a­nal­y­sis at new 2023 in­sert - one year lat­er, which em­pha­sized the re­lief of the gold­en spi­ral with φ each 2­/5 of wrap (144°) as prob­a­bly the most sig­nif­i­cant, even com­pared to the clas­sic one (90%).

At this point, all that was mis­sing was a last stage and an in­cen­tive for re­search, which would al­low en­thu­si­asts to di­rect­ly ver­i­fy the ac­tu­al gold­en de­vel­op­ment on their im­ag­es, a tool like the pre­vi­ous ac­ces­si­ble on­line and free, with re­fined add­ed fea­tures, up to the for­mu­la­tion of the anti-spi­ral, and the per­spec­tive ad­ap­ta­tion of the paths.
And here it is.