The half-regular Octagon in the Knauth-Figure
by Klaus-Anton, Jan 26, 2013, 11:19 AM
Here http://www.artofproblemsolving.com/blog/79379 you have seen, how the Knauth-Figure has jumped out of the Egyptian Triangle, which is a rectangular triangle and which has the sidelength (3,4,5).
To have seen this, this gives up to think. It makes curious to know a little bit more about the Knauth-Figure. One really wants to have a nearer look. It can be constructed using the Knauth-Triangle, which has the sidelength (1, sqrt(5)/2, sqrt(5)/2), which is easily found in the unit-square. Here now I show you the Knauth-Figure itself directly:
![[asy]
//import graph; // graph.asy already is loaded by AoPS-asy.exe.
size(10cm);
pair A1, A2, A3, A4;
pair B1, B2, B3, B4;
pair C1, C2, C3, C4, C5, C6, C7, C8;
pair D1, D2, D3, D4, D5, D6, D7, D8;
pair M=(.5,.5); label("$M$", M);
A1=(0,0); label("$A_1$", A1, SW);
A2=(1,0); label("$A_2$", A2, SE);
A3=(1,1); label("$A_3$", A3, NE);
A4=(0,1); label("$A_4$", A4, NW);
B1=(.5,0); label("$B_1$", B1, S);
B2=(1,.5); label("$B_2$", B2, E);
B3=(.5,1); label("$B_3$", B3, N);
B4=(0,.5); label("$B_4$", B4, W);
C1=intersectionpoint(A1--B3, A2--B4);
label("$C_1$", C1, 1.1*dir(M--C1));
C2=intersectionpoint(B1--A4, A1--B2);
label("$C_2$", C2, 1.2*dir(M--C2));
C3=intersectionpoint(B1--A3, A2--B4);//M--C3=0.5
label("$C_3$", C3, dir(M--C3));
C4=intersectionpoint(A2--B3, A1--B2);
label("$C_4$", C4, dir(M--C4));
C5=intersectionpoint(B1--A3, B2--A4);
label("$C_5$", C5, dir(M--C5));
C6=intersectionpoint(A2--B3, A3--B4);
label("$C_6$", C6, 1.6*dir(M--C6));
C7=intersectionpoint(A1--B3, A4--B2);
label("$C_7$", C7, 1.2*dir(M--C7));
C8=intersectionpoint(B1--A4, B4--A3);
label("$C_8$", C8, 1.1*dir(M--C8));
D1=intersectionpoint(B1--A4, B4--A2);
label("$D_1$", D1, dir(D1--M));
D2=midpoint(A1--B2); label("$D_2$", D2, S);
D3=intersectionpoint(A1--B2, B1--A3);
label("$D_3$", D3, dir(D3--M));
D4=midpoint(B1--A3); label("$D_4$", D4, .8*E);
D5=intersectionpoint(A2--B3, B2--A4);
label("$D_5$", D5, dir(D5--M));
D6=midpoint(A3--B4); label("$D_6$", D6, N);
D7=intersectionpoint(A1--B3, A3--B4);
label("$D_7$", D7, dir(D7--M));
D8=midpoint(B1--A4); label("$D_8$", D8, .8*W);
draw(A1--A2--A3--A4--cycle, dotted);
draw(A1--B3--A2, blue);
draw(A4--B1--A3, blue);
draw(A1--B2--A4, red);
draw(A2--B4--A3, red);
draw(Circle(M, arclength(M--C1)), red);
draw(Circle(M, arclength(M--D1)), orange);
draw(Circle(M, arclength(M--D2)), green);
[/asy]](//latex.artofproblemsolving.com/a/3/4/a34587858d30a8035640e2d76fc5a021228579da.png)
Oh, what a beautiful eight-star. It shows up so much relations - more or less hidden - and now one understands better, that there are so many people, who have spent time and time, to try to know "it" exactly, what everything is going on there.
One frequently asked question is, wheter the octagon in the the center of the Knauth-Figure is regular or not. So far it is this, as all sidelength of this octagon are equal. But - if you beleive me this, the angles of the octagon cannot be equal, they must be oszilating.
Here http://www.schulen-frauenfeld.ch/cm_data/public/loesungen_6b_e.pdf (called 26.01.2013, secondly 01.02.2013, here now with red page-numbering page 136, and with black page-numbering page 162) on page 136 is stated not only all the sidelenths and but also too all the angles are equal. This is asserted there with the words: "Im Inneren entsteht ein regelmässiges Achteck." But asy.exe has helped me to be able to show that this is not true so. Here I have good reason to "believe" asy.exe, which I have asked to make this visible by drawing the orange and the green circles.
For now, this is enough. Later we will look, how long the sidelengths of the octagon are exactly and what are the sizes of the here-in appearing angles.
To have seen this, this gives up to think. It makes curious to know a little bit more about the Knauth-Figure. One really wants to have a nearer look. It can be constructed using the Knauth-Triangle, which has the sidelength (1, sqrt(5)/2, sqrt(5)/2), which is easily found in the unit-square. Here now I show you the Knauth-Figure itself directly:
![[asy]
//import graph; // graph.asy already is loaded by AoPS-asy.exe.
size(10cm);
pair A1, A2, A3, A4;
pair B1, B2, B3, B4;
pair C1, C2, C3, C4, C5, C6, C7, C8;
pair D1, D2, D3, D4, D5, D6, D7, D8;
pair M=(.5,.5); label("$M$", M);
A1=(0,0); label("$A_1$", A1, SW);
A2=(1,0); label("$A_2$", A2, SE);
A3=(1,1); label("$A_3$", A3, NE);
A4=(0,1); label("$A_4$", A4, NW);
B1=(.5,0); label("$B_1$", B1, S);
B2=(1,.5); label("$B_2$", B2, E);
B3=(.5,1); label("$B_3$", B3, N);
B4=(0,.5); label("$B_4$", B4, W);
C1=intersectionpoint(A1--B3, A2--B4);
label("$C_1$", C1, 1.1*dir(M--C1));
C2=intersectionpoint(B1--A4, A1--B2);
label("$C_2$", C2, 1.2*dir(M--C2));
C3=intersectionpoint(B1--A3, A2--B4);//M--C3=0.5
label("$C_3$", C3, dir(M--C3));
C4=intersectionpoint(A2--B3, A1--B2);
label("$C_4$", C4, dir(M--C4));
C5=intersectionpoint(B1--A3, B2--A4);
label("$C_5$", C5, dir(M--C5));
C6=intersectionpoint(A2--B3, A3--B4);
label("$C_6$", C6, 1.6*dir(M--C6));
C7=intersectionpoint(A1--B3, A4--B2);
label("$C_7$", C7, 1.2*dir(M--C7));
C8=intersectionpoint(B1--A4, B4--A3);
label("$C_8$", C8, 1.1*dir(M--C8));
D1=intersectionpoint(B1--A4, B4--A2);
label("$D_1$", D1, dir(D1--M));
D2=midpoint(A1--B2); label("$D_2$", D2, S);
D3=intersectionpoint(A1--B2, B1--A3);
label("$D_3$", D3, dir(D3--M));
D4=midpoint(B1--A3); label("$D_4$", D4, .8*E);
D5=intersectionpoint(A2--B3, B2--A4);
label("$D_5$", D5, dir(D5--M));
D6=midpoint(A3--B4); label("$D_6$", D6, N);
D7=intersectionpoint(A1--B3, A3--B4);
label("$D_7$", D7, dir(D7--M));
D8=midpoint(B1--A4); label("$D_8$", D8, .8*W);
draw(A1--A2--A3--A4--cycle, dotted);
draw(A1--B3--A2, blue);
draw(A4--B1--A3, blue);
draw(A1--B2--A4, red);
draw(A2--B4--A3, red);
draw(Circle(M, arclength(M--C1)), red);
draw(Circle(M, arclength(M--D1)), orange);
draw(Circle(M, arclength(M--D2)), green);
[/asy]](http://latex.artofproblemsolving.com/a/3/4/a34587858d30a8035640e2d76fc5a021228579da.png)
Oh, what a beautiful eight-star. It shows up so much relations - more or less hidden - and now one understands better, that there are so many people, who have spent time and time, to try to know "it" exactly, what everything is going on there.
One frequently asked question is, wheter the octagon in the the center of the Knauth-Figure is regular or not. So far it is this, as all sidelength of this octagon are equal. But - if you beleive me this, the angles of the octagon cannot be equal, they must be oszilating.
Here http://www.schulen-frauenfeld.ch/cm_data/public/loesungen_6b_e.pdf (called 26.01.2013, secondly 01.02.2013, here now with red page-numbering page 136, and with black page-numbering page 162) on page 136 is stated not only all the sidelenths and but also too all the angles are equal. This is asserted there with the words: "Im Inneren entsteht ein regelmässiges Achteck." But asy.exe has helped me to be able to show that this is not true so. Here I have good reason to "believe" asy.exe, which I have asked to make this visible by drawing the orange and the green circles.
For now, this is enough. Later we will look, how long the sidelengths of the octagon are exactly and what are the sizes of the here-in appearing angles.
This post has been edited 5 times. Last edited by Klaus-Anton, Feb 8, 2013, 1:52 PM
July 2025
June 2025