Circle Radius of Trapezoid

by Klaus-Anton, Jun 4, 2025, 5:58 PM

You find the question on fb Beautiful Geometry Problems CIRCLE RADIUS = ???

I have made me this picture and wanted asy to display me the degrees of $O--T$ .
$[asy] usepackage("nicefrac"); pair A,B,C,D,O,T; real real_r=10/3; A=(-real_r,1*real_r); D=(-real_r,-real_r); B=A+5; C=D+10; draw(A--B--C--D--cycle); draw(circle(O,real_r)); draw((0,-real_r)--O--(real_r,0)); label("$A$",A,NW); label("$B$",B,dir(D--B)); label("$C$",C,dir(A--C)); label("$D$",D,SW); label("$O$",O,dir(C--B)*dir(90)); T=real_r*dir(C--B)*dir(-90); label("$T$",T, dir(O--T)); label("$\nicefrac{10}{3}$", O--(real_r,0)); draw(O--T); dot(O); real my_degrees=degrees(dir(C--B)*dir(-90)); dot(real_r*dir(C--B)*dir(-90)); dot(dir(my_degrees),red); dot(A^^B^^C^^D); label("$5$", A--B,N); label("$10$", C--D,S); label(format("degrees(dir$(O--T)$)$\,=%.6f\dots^\circ$", my_degrees), midpoint(D--C), 5*S); import geometry; perpfactor=.8; perpendicularmark(A,SE); perpendicularmark(D,NE); shipout(bbox(2mm, FillDraw(white,white))); [/asy]$

trapezoid

radius

Construct a Triangle knowing the Lengths of 3 Sides

by Klaus-Anton, Jun 1, 2025, 1:20 PM

You find this in How to construct the point S of this tetrahedron?

$[asy] // http://asymptote.ualberta.ca/ pair barycentric(pair A=(0,0), pair B=(0,0), real a=1, real b=0){ return (a*A+b*B)/(a+b);} // Application: construct a triangle knowing the lengths of 3 sides unitsize(1cm); real a=6, b=5, c=2.5; pair B=(0,0), C=(a,0); pair H=barycentric(B,C,1/(a^2+c^2-b^2),1/(a^2+b^2-c^2)); real bt=abs(H-B); real h=sqrt(c*c-bt*bt); pair A=H+h*dir(90); draw(box(H,H+(.2,.2)),red); draw(A--H,red); draw(A--B--C--cycle); label("$A$",A,plain.N); label("$B$",B,plain.SW); label("$C$",C,plain.SE); label("$H$",H,plain.S); shipout(bbox(2mm,FillDraw(white,white))); [/asy]$

Note: You can find an incomplete asy as bary.asy of vEnhance / dotfiles.

This post has been edited 3 times. Last edited by Klaus-Anton, Jun 1, 2025, 1:23 PM

bary.asy

barycenter

barycentric

A Minipage as Explanation

by Klaus-Anton, Jun 1, 2025, 1:07 PM

Black Mild (2021) wrote:

Black Mild in Circle made from intersecting spheres (modified)
$[asy] // http://asymptote.ualberta.ca/ usepackage("ragged2e"); // for justify command usepackage("amsmath"); unitsize(7mm); real r=4.2, s=3; // two radii real c=5; // distance of two centers pair A=(0,0), B=c*dir(0); //two centers pair M=intersectionpoints(circle(A,r),circle(B,s))[0]; real dAH=(c^2+r^2-s^2)/(2*c); real h=sqrt(r^2-dAH^2); pair H=A+dAH*unit(B-A); draw(Label("$h$",align=W),M--H,magenta); draw(Label("$r$",align=NW),A--M,red); draw(Label("$s$",align=NE),B--M,blue); draw(circle(A,r),red); draw(circle(B,s),blue); label("$M$",M+.5dir(80),magenta); label("$H$",H,S,magenta); draw(Label("$c$"),A--B); dot("$A$",A,SW); dot("$B$",B,SE); dot(H^^M,magenta); string explanation=minipage("\justify{ By the law of cosines for the triangle $AMH$, we have $$AH=r \cos A=\dfrac{c^2+r^2-s^2}{2c}.$$ Hence, the radius of the intersecting circle is $$h=\sqrt{r^2-AH^2}.$$ }",7cm); label(explanation,point(S)+(1,-5),Fill(3mm,lightyellow)); shipout(scale(.7)*bbox(5mm,FillDraw(white,white))); [/asy]$

This post has been edited 1 time. Last edited by Klaus-Anton, Jun 1, 2025, 1:08 PM

Equilateral triangle

by Klaus-Anton, Jun 1, 2025, 12:01 PM

$[asy] import geometry; real real1=90; real real2=30; pair A=dir(real1) ,B=dir(2*real1+real2) ,C=dir(-real2) ,O=(0,0) ,D=dir(real2) , myE=(C.x,D.y+1) ,F=intersectionpoint(line(B,D),line(A,C)); ; triangle t=triangle(A,B,C); draw(t); label("$A$",A,dir(C--A)); label("$B$",B,dir(O--B)); label("$C$",C,dir(A--C)); perpfactor=.8; //perpendicularmark(A, dir(A--myE,A--C)); perpendicularmark(A, dir(A--myE)*dir(-45)); perpendicularmark(F, dir(F--O)*dir(45)); perpendicularmark(C, dir(NE)); draw(B--D,red); draw(circle(O,1)); draw(C--D--(C.x,D.y+1)--A); draw(circle(D,arclength(C--D)),blue); dot(A^^B^^C,FillDraw(white,black)); dot("$O$",O,dir(A--B)*dir(.5*30)); dot("$D$",D,dir(0+15*1.5)); dot("$E$",myE,N); dot("$F$",F,1.2*dir(C--A,O--D)*dir(-10)); label(rotate(real2)*"$1$",relpoint(O--B,.4),.8*dir(B--O)*dir(real1),white); // https://github.com/vEnhance/dotfiles/blob/main/asy/olympiad.asy add(pathticks(A--B,r=.5,s=4,white)); add(pathticks(B--C,r=.5,s=4,white)); add(pathticks(C--D,2,r=.5,spacing=3,s=4,white)); add(pathticks(D--myE,2,r=.5,spacing=3,s=4,white)); //draw(circle(C,sqrt(3)));// A--B=A--C=B--C=sqrt(3) shipout(bbox(2mm,FillDraw(white+red,white))); [/asy]$

Equilateral Triangle

How does a new line change that?

by Klaus-Anton, May 4, 2025, 1:39 PM

It has been asked:

Pistons wrote:

How does a new line change that?

(Pistons Different Image - And here read my reply there).

But Pistons there had not read exactly. Compare:

aidan0626 wrote:

probably an issue where the aops servers cached the wrong image
like if you remove a new line, you get this
$[asy] unitsize(1.5 cm); pair A, B, C, D, O, T; O = (0,0); T = dir(20); B = extension(T, T + rotate(90)*(T), (0,1), (1,1)); C = extension(T, T + rotate(90)*(T), (0,-1), (1,-1)); A = reflect((0,0),(0,1))*(B); D = reflect((0,0),(0,1))*(C); draw(A--B--C--D--cycle); draw(Circle(O,1)); label("$A$", A, NW); label("$B$", B, NE); label("$C$", C, SE); label("$D$", D, SW); [/asy]$

You see, aidan0626 told him not to add a new line. He told him to remove a new line. Okay. Also there the other way round would either work as well as i think.

In my eyes this is a similar case as i had explained it here:

Klaus-Anton in reply to sirinivi in how to remove y from diagram

Recently to me too happened a similar case, which has made me wondering not little.

In my reply in writing problem asymptote (post #10) appear two line segments of an earlier version i had made in TeXeR. The statement $\textrm{draw(B-{}-{-A}-{}--C);}$ does not stand there in the code. But in spite of this the two segments are visible. Strange enough. These two segments there are visible. But the code does not say to asy to draw these segments.

Let us see:

$[asy] //unitsize(3.1cm); size(8cm); pair A, B, C, P, P0, Q; A = (1,0); B = (0,1); C = (0,0); P = extension(C, C + dir(16), A, B); P0 = (-P.y,P.x); pair Pprime = (-P.y,P.x); Q = extension(C, C + dir(16 + 45), A, B); pair Q61=dir(61); pair P16_plus_90=dir(16+90); draw(A--B--C--cycle); draw(C--P); draw(C--P0); draw(P--P0,dashed); draw(C--Q); draw(P--dir(16) ^^Q--Q61 ^^Pprime--dir(16+90) ^^arc(C,1,0,16+90) ,dotted); dot(dir(16),UnFill); dot(dir(16+90),UnFill); dot(Q61,UnFill); import geometry; perpfactor=.8; perpendicularmark(C, dir(Q),red); perpendicularmark(C, dir(NE),blue); label("$A$", A, SE); label("$B$", B, N); label("$C$", C, SW); label("$P$", P, dir(P--Q,C--P)*dir(-15)); label("$P'$", Pprime, dir(C--Pprime)*dir(90)); real angleQCP=45, anglePCA=16, angleBCA=90, angleP16_plus_90CA=16+90 ; //label(rotate(-90+8,dir(8))*"$16^\circ$",dir(8), dir(8)); label(//rotate(-90+8,dir(8))* "$y$",dir(8), dir(8)); label(rotate(-90+22.5+16,dir(16+22.5))*"$45^\circ$",dir((16+22.5)), dir((16+45)/2)); label("$Q$", Q, 1.25*dir(C--Q,Q--P)*dir(-3.5)); //label(rotate(8)*"$16^\circ$",arc(C,1,90,90+16)); label(rotate(8)*"$y$",arc(C,1,90,90+16)); //label(rotate(-29/2)*"$29^\circ$",arc(C,1,16+45,90)); label(rotate(-29/2)*"$x$",arc(C,1,16+45,90)); //draw(rotate(-90+(90+16)/2)*"$90^\circ+16^\circ$",arc(C,1.25,0,90+16),Arrow);//autosizes draw(rotate(-90+(90+16)/2)*"$90^\circ+y$",arc(C,1.25,0,90+16),Arrow);//autosizes shipout(bbox(2mm,FillDraw(white,white))); [/asy]$

Fig. 1: segments of $\textrm{draw(B-{}-{-A}-{}--C);}$ visible

$[asy] //unitsize(3.1cm); size(8cm); pair A, B, C, P, P0, Q; A = (1,0); B = (0,1); C = (0,0); P = extension(C, C + dir(16), A, B); P0 = (-P.y,P.x); pair Pprime = (-P.y,P.x); Q = extension(C, C + dir(16 + 45), A, B); pair Q61=dir(61); pair P16_plus_90=dir(16+90); draw(A--B--C--cycle); draw(C--P); draw(C--P0); draw(P--P0,dashed); draw(C--Q); draw(P--dir(16) ^^Q--Q61 ^^Pprime--dir(16+90) ^^arc(C,1,0,16+90) ,dotted); dot(dir(16),UnFill); dot(dir(16+90),UnFill); dot(Q61,UnFill); import geometry; perpfactor=.8; perpendicularmark(C, dir(Q),red); perpendicularmark(C, dir(NE),blue); label("$A$", A, SE); label("$B$", B, N); label("$C$", C, SW); label("$P$", P, dir(P--Q,C--P)*dir(-15)); label("$P'$", Pprime, dir(C--Pprime)*dir(90)); real angleQCP=45, anglePCA=16, angleBCA=90, angleP16_plus_90CA=16+90 ; //label(rotate(-90+8,dir(8))*"$16^\circ$",dir(8), dir(8)); label(//rotate(-90+8,dir(8))* "$y$",dir(8), dir(8)); label(rotate(-90+22.5+16,dir(16+22.5))*"$45^\circ$",dir((16+22.5)), dir((16+45)/2)); label("$Q$", Q, 1.25*dir(C--Q,Q--P)*dir(-3.5)); //label(rotate(8)*"$16^\circ$",arc(C,1,90,90+16)); label(rotate(8)*"$y$",arc(C,1,90,90+16)); //label(rotate(-29/2)*"$29^\circ$",arc(C,1,16+45,90)); label(rotate(-29/2)*"$x$",arc(C,1,16+45,90)); //draw(rotate(-90+(90+16)/2)*"$90^\circ+16^\circ$",arc(C,1.25,0,90+16),Arrow);//autosizes draw(rotate(-90+(90+16)/2)*"$90^\circ+y$",arc(C,1.25,0,90+16),Arrow);//autosizes shipout(bbox(2mm,FillDraw(white,white))); [/asy]$

Fig. 2: segments of $\textrm{draw(B-{}-{-A}-{}--C);}$ un-visible

And this is even so as it is. - Neither in the code for Fig. 1 or in the code for Fig. 2 you can find the statement $\textrm{draw(B-{}-{-A}-{}--C);}$ . So it is.

This post has been edited 5 times. Last edited by Klaus-Anton, May 5, 2025, 8:42 AM

iced tea

by Klaus-Anton, Apr 25, 2025, 12:50 PM

Did you see my reply on having a "rainbow" color?

You could add there StackExchange Rainbow text on a surface in Asymptote, and - as a nice modification - for example this too:

$[asy] // modified from https://tex.stackexchange.com/questions/430142/rainbow-text-on-a-surface-in-asymptote size(5cm); pen[][] p={{white,grey,black}, {red,green,blue}, {cyan,magenta,yellow}}; latticeshade(texpath("iced tea"),p); shipout(defaultfilename,bbox(2mm,RadialShade(mediumred+white,red))); [/asy]$

$[asy] // modified from https://tex.stackexchange.com/questions/430142/rainbow-text-on-a-surface-in-asymptote size(5cm); pen[][] p={{white,grey,black}, {red,green,blue}, {cyan,magenta,yellow}}; latticeshade(texpath("Klaus"),p); shipout(bbox(2mm,RadialShade(mediumred+white,red)) ); shipout(Landscape); [/asy]$

$[asy] //modified from https://tex.stackexchange.com/questions/430142/rainbow-text-on-a-surface-in-asymptote size(5cm); pen[][] p={{white,grey,black}, {red,green,blue}, {cyan,magenta,yellow}}; latticeshade(texpath(yscale(2)*"BlackShark28"),p); shipout(rotate(-6)*bbox(2mm,RadialShade(mediumred+white,red)) ); shipout(rotate(0)*bbox(2mm,FillDraw(red,red))); [/asy]$

$[asy] //modified from https://tex.stackexchange.com/questions/430142/rainbow-text-on-a-surface-in-asymptote size(5cm); pen[][] p={{white,grey,black}, {red,green,blue}, {cyan,magenta,yellow}}; latticeshade(texpath(scale(3)*(rotate(15))*"\textbf{\textit{Ela Pan}}"),p); shipout(rotate(-6)*bbox(2mm,RadialShade(mediumred+white,red)) ); shipout(rotate(0)*bbox(2mm,FillDraw(red,red))); [/asy]$

$[asy] //modified from https://tex.stackexchange.com/questions/430142/rainbow-text-on-a-surface-in-asymptote size(5cm); //defaultpen(fontsize(10pt)+TimesRoman()); pen[][] p={{white,grey,black}, //{red,green,blue}, {orange,green,white}, {cyan,magenta,yellow}}; latticeshade(texpath(scale(3)*(rotate(15))*"\textbf{\textit{Larysa Dbg}}"),p); shipout(rotate(6)*bbox(2mm,RadialShade(mediumred+white,red)) ); shipout(rotate(-30)*bbox(2mm,FillDraw(red,red))); shipout(rotate(90)*bbox(0mm,FillDraw(red,red))); [/asy]$

This post has been edited 6 times. Last edited by Klaus-Anton, Jul 5, 2025, 3:47 PM

Rainbow

latticeshade

RadialShade

Birthday asymptote

by Klaus-Anton, Mar 25, 2025, 4:20 PM

I answerd to RoyalPrince Birthday asymptote (and RoyalPrince he directly prompted with "Thank you!"):

$[asy] /* http://asymptote.ualberta.ca/ click pdf click download open with Adobe Acrobat Reader ctr-h and after that ctr-l ===> fullscreen */ usepackage("calligra"); tex("\DeclareFontFamily{T1}{calligra}{}"); tex("\DeclareFontShape{T1}{calligra}{m}{n}{<->s*[1.55]callig15}{}"); settings.render=4; unitsize(1.75cm); real gr=(sqrt(5)-1)/2; pair A=(.5,0), B=(1.5,0); real ra=1+(2*.0675), rb=1.125; radialshade(//currentpicture box((-1.125,-1.375),(3.0,1.5))//,stroke=false //default ,blue+white, A, ra, extenda=false// true is default ,red+white,B,rb,extendb=false);// true is default label(scale(.9)*slant(-gr^.5)*"\huge{\textcalligra{Happy Birtday, Sister!}}",(0,0),yellow+white+.5orange); shipout(defaultfilename,bbox(.5mm,RadialShade(paleblue,red))); [/asy]$
This is a small modification from Figure 6 in my contribution radialshade

This post has been edited 2 times. Last edited by Klaus-Anton, Mar 25, 2025, 4:34 PM

Isosceles Trapezoid with Incircle

by Klaus-Anton, Mar 25, 2025, 3:58 PM

I pick up here one of my replies in Asy Help.
$[asy] size(7cm); pair A,B,C,D; real r = (3+sqrt(5))/2; real s = sqrt(r); A = (-r,0); B = (r,0); C = (1,2*s); D = (-1,2*s); draw(A--B--C--D--cycle); pair O = (0,s); draw(shift(O)*scale(s)*unitcircle); dot(O); pair X,Y; X = (0,0); Y = (0,2*s); draw(X--Y); //pair P = 0.73*C+0.27*B; pair P=relpoint(B--C, r/(r+1)); import geometry; //draw(line(O,P)); perpfactor=.8; perpendicularmark(P, dir(P--O,P--B),red); draw(O--P); dot(P); label("$1$",(C+P)/2,NE); label("$r$",(B+P)/2,NE); //label("$r-1$",(X+B)/2,S); label("$r-1$", midpoint((C.x,0)--(B.x,0)),S,blue); label("$1$",(Y+C)/2,N); label("$s$",(O+Y)/2,W); label("$s$",(O+X)/2,W); draw(B--C--(1,0)--cycle,blue+1bp); label("$A$",A, -dir(A--B,A--D)); label("$B$",B, -dir(B--A,B--C)); label("$C$",C, -dir(C--B,C--D)); label("$D$",D, -dir(D--A,D--C)); label("$O$",O,dir(O--intersectionpoint(line(D,A),line(O,P)),O--O-1)); //dot(intersectionpoint(line(D,A),line(O,P))); label("$P$",P,dir(O--P)); label(format("$\texttt{degrees(dir(O--P))}=%.3f\dots^\circ$",degrees(dir(O--P))),.75S); shipout(bbox(2mm,FillDraw(white,white))); [/asy]$

This post has been edited 2 times. Last edited by Klaus-Anton, Mar 25, 2025, 4:28 PM

How do i get a fontsize larger than 25pt in asy?

by Klaus-Anton, Mar 11, 2025, 9:31 AM

In LABELS - Effets de texte of gmarris in ASYMPTOTE un puissant langage de graphisme vectoriel Galeries de 724 exemples, par thèmes, de figures réalisées avec Asymptote there you find an example which demonstrates that asy here does not really hear on the programming code. It ought draw too in a fontsize bigger than 25 pt, but does not do so.

You remember. $10pt\times1.2^5=24.8832pt\approx25pt$ . This is a well known limit in $\LaTeX$ . But there are ways to make oversteps over this limit.

Okay.

$[asy] unitsize(1cm); import math; defaultpen(fontsize(10pt)); label("$10$",(0,0),blue); label("$15$",(1,0),fontsize(15pt)+red); label("$20$",(2,0),fontsize(20pt)+green); label("$25$",(3,0),fontsize(25pt)+blue); label("$30$",(4,0),fontsize(30pt)+red); label("$35$",(5,0),fontsize(35pt)+green); label("$40$",(6,0),fontsize(40pt)+blue); label("$45$",(7,0),fontsize(45pt)+red); label("Comprendre la taille des \'etiquettes (labels).",(0,1),E); label("Comparer avec l'exemple suivant !",(7.5,-1),W); shipout(bbox(2mm, FillDraw(white,white))); [/asy]$

Figure 0007: fig_ab01_030420_fontsize.asy (gmerris)

$[asy] unitsize(1cm); import math; /////////////////////////////////////////////////////////////////// import fontsize; // <<<<<<<<< pour obtenir des tailles plus grandes /////////////////////////////////////////////////////////////////// defaultpen(fontsize(10pt)); label("$10$",(0,0),blue); label("$15$",(1,0),fontsize(15pt)+red); label("$20$",(2,0),fontsize(20pt)+green); label("$25$",(3,0),fontsize(25pt)+blue); label("$30$",(4,0),fontsize(30pt)+red); label("$35$",(5,0),fontsize(35pt)+green); label("$40$",(6,0),fontsize(40pt)+blue); label("$45$",(7,0),fontsize(45pt)+red); label("Comprendre la taille des \'etiquettes (labels).",(0,1),E); label("Comparer avec l'exemple pr\'ec\'edent !",(7.5,-1),W); shipout(bbox(2mm, FillDraw(white,white))); [/asy]$

Figure 0008: fig_ab01_030420_fontsize_package.asy (gmerris)

Figure 0008 realizes it that it can draw the fonts bigger than in 25 pt by the additional line in the asy programming code: import fontsize;

LaTeX

Asymptote

fontsize

Beitrag zur Diskussion der Orientierung der Rose im Wappen von NRW (BRD)

by Klaus-Anton, Mar 7, 2025, 9:18 AM

Guck dir das doch mal an in dem Artikel: wiki Wappen_Nordrhein-Westfalens.

In den dortigen Quellen findet sich GiW1996_1/GiW_1996_1_NAGEL_LANDESWAPPEN.pdf. Auf Seite 39 f. schreibt Nagel: „Man hat verschiedentlich davon gesprochen, daß die lippische Rose im Landeswappen auf dem Kopfe stehe. Ich gestehe, daß ich mir schwer vorstellen kann, wie eine Rosenblüte auf dem Kopf stehen soll.“

Der wiki-Artikel schreibt:

Quote:

„Das untere silberne Feld mit der eingebogenen Spitze stellt die Lippische Rose dar, das dynastische Symbol des Hauses Lippe, das bereits in einem Siegel Hermanns II. von Lippe aus dem Jahre 1218 nachgewiesen ist.[8] Im Landeswappen steht die Rose für das Land Lippe, das nach dem Zweiten Weltkrieg seine Selbstständigkeit als Freistaat aufgeben musste und am 21. Januar 1947 in das Land Nordrhein-Westfalen eingegliedert wurde. Im Staatswappen von Lippe wie auch in den meisten kommunalen Wappen von Lippe zeigt eines der fünf goldenen Kelchblätter gerade nach unten. Gegenüber diesen Darstellungen wurde die lippische Rose im nordrhein-westfälischen Landeswappen um 36 Grad gedreht, so dass eines der goldenen Kelchblätter gerade nach oben zeigt.“

Und dann heißt es in der Bild-Unterschrift zum Staatswappen Lippes: Die „korrekt“ orientierte Lippische Rose im Staatswappen Lippes.

Warum obig zitierter Nagel seine Einsichtsschrwierigkeit genau hat, ist nicht richtig zu verstehen.

In wiki - Drudenfuß (Heraldik) wird das Gemeinte des Auf-dem-Kopf-Stehens deutlicher, da heißt es z.B.: „Der Begriff Drudenfuß wird in der Heraldik für alle Pentagramme verwendet, egal ob die Figur mit der Spitze nach oben oder unten zeigt.“

Ein Bannzeichen des Bösen? In Werner Robl Das Pentagramm in Barock und Klassizismus heißt es zu Goethe:

Quote:

„Im Sommer 1830 entstand davor ein Bodenmosaik aus großen Kalk- und Kieselsteinen, das als Bannzeichen des Bösen ein Pentagramm mit Um- und Inkreis darstellte. Trotz der in Faust 1 berichteten Fragilität hatte es für Goethe und seine Gäste noch jenen Schutz und jene Sicherheit zu symbolisieren, den es über Jahrtausende, von der Vor- bis zur Neuzeit, repräsentiert hatte.“

Sieht man sich das Bild (goethe.jpg) an, so scheint eine Ausrichtung bezüglich der Wege und zum Haus hin mit angedacht gewesen zu sein.

Und übrigens. - Hast du das gesehen: Klaus-Anton's blog 2025 The so called Number of the beast: chi, xi, stigma (666)?

Ach! - Und übrigens auch. Schneide doch mal einen Apfel horzontal durch. Was erscheint dir?

Mario Livio 2002. The Golden Ratio. The Story of Phi , the World's Most Astonishing Number. Broadway Books. New York. wrote:

Take, for example, an ordinary apple, the fruit often associated (probably mistakenly) with the tree of knowledge that figures so prominently in the biblical account of humankind’s fall from grace, and cut it through its girth. You will find that the apple’s seeds are arranged in a five-pointed star pattern, or pentagram (Figure 3). Each of the five isosceles triangles that make the corners of a entagram has the property that the ratio of the length of its longer side to the shorter one (the implied base) is equal to the Golden Ratio, 1.618…. But, you may think, maybe this is not so surprising. After all, since the Golden Ratio has been defined as a geometrical proportion, perhaps we should not be too astonished to discover that this proportion is found in some geometrical shapes.

(Page 17)

Das mit dem Apfel, das verwundert eigentlich nicht sehr. Die Büte ist im Grundaufbau fünfzählig. Fünf Blütenblätter und fünf Kelchblätter. Hm. Aber. Na ja. Ja, ja. - Ja, doch. Genau. So wie die Rose selbst gehört ja auch der Apfel botanisch gesehen (eben so wie die Birne und viele andere nehr) zu den Rosacaen, den Rosengewächsen. Aha. Ach so! - Ja, ja. So ist das.

This post has been edited 3 times. Last edited by Klaus-Anton, Mar 9, 2025, 10:44 AM

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by Neurotricity, Oct 20, 2022, 9:38 AM
Can I be contrib? Thanks!
by MC413551, Oct 8, 2022, 2:14 PM
Very nice blog
by programmeruser, Sep 19, 2022, 12:23 PM
Wow! this is EPIC!
by mathkid369, Jul 11, 2022, 2:43 PM
This blog is amazing! I've found so much useful information on here. Thank you!
by je100, Mar 25, 2022, 5:04 PM
Unarguably one of the best blogs on AoPS created by one of the most amazing people on AoPS
by player01, Mar 24, 2022, 6:38 PM
Thank you for this wonderful resource, K-A! It's helped me a lot. Not many sites have asymptotic tutorials that make sense, or Google just understands "asymptote" wrong.
by fuzimiao2013, Dec 12, 2021, 6:43 PM
wow you are a pro at asymptote great job klaus-anton!!!
by HWenslawski, Oct 20, 2021, 5:24 PM
Amazing blog. You are so good at asymptote!
by player01, Aug 27, 2021, 12:48 PM
ngl, Klaus-Anton is pr0 at LaTeX and asymptote
by mathlearner2357, Mar 19, 2021, 9:44 AM
Thanks! This blog had a lot of asymptote functions that I needed for olympiad geometry diagrams.
by eduD_looC, Mar 16, 2021, 12:28 AM
@player01 now 30k visits
by ObjectZ, Feb 2, 2021, 2:14 PM
@pog now 29k visits
by player01, Nov 11, 2020, 5:14 PM
18 shouts and 26k visits?
by pog, Sep 9, 2020, 3:48 PM
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by ARay10, Sep 4, 2020, 4:21 PM
What software do you use to make asymptote diagrams?

-πφ
by piphi, Jun 18, 2020, 3:09 AM
dang this is an old blog
by CoolCarsOnTheRun, Jun 7, 2020, 10:07 PM
Is it possible to export an asymptote drawing as an SVG?

-piphi
by piphi, May 12, 2020, 6:56 PM
Yah, I remember that...
by sonone, May 1, 2020, 2:39 PM
Actually, the making pies thing is from AoPS challenge problems in PA2.
by RedSox2, Apr 23, 2020, 4:53 PM
I like your blog. How did you learn so much?
by sonone, Apr 3, 2020, 2:36 PM
nice blog!
by matharcher, Mar 21, 2020, 2:10 AM
wow, this is awesome! Im subscribing
by piphi, Feb 12, 2020, 8:28 PM
Yes sure it is!
by mathboy282, Dec 21, 2019, 9:38 PM
I like this blog. It is very helpful.
by Chishimotoji, Jan 23, 2019, 3:12 AM
Great blog!
by WizardMath, Mar 9, 2018, 10:59 AM
Asymptote = good = probably better that I ever will be
by coolmath34, Nov 23, 2017, 1:03 AM
Wow you are good at asy!
by fuegocaliente, May 24, 2017, 9:20 PM
first shout of 2017!!
by Mathisfun04, Mar 6, 2017, 3:00 PM
Nice asymptote!
by Boomer, Apr 2, 2014, 4:22 PM
I wonder, I wonder.
by Klaus-Anton, Jan 8, 2011, 6:19 PM
First shout!
by El_Ectric, Jan 4, 2011, 9:14 PM

38 shouts