Character Tables for Point Goups used in Chemistry

Character tables are an important tool in many parts of molecular chemistry, particularly in spectroscopy. On this page, you can find character tables for all remotely interesting discrete axial point groups, plus the groups for cubic and icosahedral symmetry.

Some point groups have irreducible representations with complex characters. These have been eliminated by creating reducible representations with real-valued characters, as is common in chemistry (applies to point groups C_n, C_nh, S_2n, T, T_h).
All point groups up to 128-fold rotations are included. This is far more than anyone is likely ever to need. To access the character tables, use the list below or enter the name of the point group into the search field (needs JavaScript).
Symmetry-adapted linear combinations of Rotations and Cartesian products up to sixth order (i functions) are listed. This is not fully implemented for icosahedral symmetry, though.

C_n

C₁

C₂

C₃

C₄

C₅

C₆

C₇

C₈

C₉

C₁₀

C₁₁

C₁₂

C₁₃

C₁₄

C₁₅

C₁₆

C₁₇

C₁₈

C₁₉

C₂₀

C₂₁

C₂₂

C₂₃

C₂₄

C₂₅

C₂₆

C₂₇

C₂₈

C₂₉

C₃₀

C₃₁

C₃₂

C_nv

C_2v

C_3v

C_4v

C_5v

C_6v

C_7v

C_8v

C_9v

C_32v

C_nh

C_s

C_2h

C_3h

C_4h

C_5h

C_6h

C_7h

C_8h

C_9h

C_32h

D_n

D₂

D₃

D₄

D₅

D₆

D₇

D₈

D₉

D₁₀

D₁₁

D₁₂

D₁₃

D₁₄

D₁₅

D₁₆

D₁₇

D₁₈

D₁₉

D₂₀

D₂₁

D₂₂

D₂₃

D₂₄

D₂₅

D₂₆

D₂₇

D₂₈

D₂₉

D₃₀

D₃₁

D₃₂

D_nh

D_2h

D_3h

D_4h

D_5h

D_6h

D_7h

D_8h

D_9h

D_32h

D_nd

D_2d

D_3d

D_4d

D_5d

D_6d

D_7d

D_8d

D_9d

D_32d

S_n

C_i

S₄

S₆

S₈

S₁₀

S₁₂

S₁₄

S₁₆

S₁₈

S₂₀

S₂₂

S₂₄

S₂₆

S₂₈

S₃₀

S₃₂

T_d

T_h

O_h

I_h

Enter Schönflies symbol:

Output explained

The output is a plain text file that has been somewhat HTML-ified; it assumes an arbitrarily wide display. It gives some standard set of information to each point group. In this section, the output is explained using D_5h as an example.

The character table of D_5h is rather short and needs no further explanation. To show the class labels and Mulliken symbols in full glory with subscripts, superscripts and Greek letters in a preformatted ASCII-like context, some tricks with JavaScript and CSS2 have to be used; users of ancient browsers will see them as C5, C5^2 and A1', A1" etc.

D5h     E       2 C5    2 C5^2  5 C2'   sh      2 S5    2 S5^3  5 sv       
A1'     1.0000  1.0000  1.0000  1.0000  1.0000  1.0000  1.0000  1.0000
A1"     1.0000  1.0000  1.0000  1.0000 -1.0000 -1.0000 -1.0000 -1.0000
A2'     1.0000  1.0000  1.0000 -1.0000  1.0000  1.0000  1.0000 -1.0000
A2"     1.0000  1.0000  1.0000 -1.0000 -1.0000 -1.0000 -1.0000  1.0000
E1'     2.0000  0.6180 -1.6180  0.0000  2.0000  0.6180 -1.6180  0.0000
E1"     2.0000  0.6180 -1.6180  0.0000 -2.0000 -0.6180  1.6180  0.0000
E2'     2.0000 -1.6180  0.6180  0.0000  2.0000 -1.6180  0.6180  0.0000
E2"     2.0000 -1.6180  0.6180  0.0000 -2.0000  1.6180 -0.6180  0.0000

Next to the character table, irreducible representations of rotations and solid harmonics are shown. This is shown in a rather compact way; a more user-friendly rendering of the same information is given below.

The output is organized in groups: The first letter triple corresponds to rotations, and the following tuples belong to real p,d,f,g,h,i basis functions (or, equivalently, Cartesian products of order 1,2,3,4,5,6). The first line (A₁^′ race) says that d₀ (z²), g₀ (z⁴), the positive linear combination (cosine component) of h_±5 and i₀ (z⁶) belong to that irreducible represention (hover over the table below to see more).

          <R> <p> <—d—> <——f——> <———g———> <————h————> <—————i—————>
A1'       ... ... ....T ....... ........T T.......... ............T
A1"       ... ... ..... ....... ......... ........... ...T.........
A2'       ..T ... ..... ....... ......... .T......... .............
A2"       ... ..T ..... ......T ......... ..........T ..T..........
E1'       ... TT. ..... ....TT. TT....... ........TT. TT..TT.......
E1"       TT. ... ..TT. ....... ......TT. ..TT....... ..........TT.
E2'       ... ... TT... TT..... ....TT... ....TT..... ........TT...
E2"       ... ... ..... ..TT... ..TT..... ......TT... ......TT.....

For cubic and icosahedral groups, however, the ordering of the basis functions within letter groups is rather arbitrary.

The D_5h character table contains noninteger characters. These are listed in the next section, along with their representation as a cosine value and (if possible) by real radicals.
```
 Irrational character values:  1.61803 = 2*cos(2*π/10) = 2*cos(π/5) = (1+√5)/2
                               0.61803 = 2*cos(4*π/10) = 2*cos(2*π/5) = (√5-1)/2
```
Nested root signs don’t display very nicely, but I am not aware of anything that would work better within the limits of HTML. See below for a full list of all values in a format more suited for programming languages.
A more explicit version of the irreducible representations of rotations and Cartesian products is shown next. The terms are given in a standard form as unnormalized real linear combinations of spherical harmonics with simplified z-dependent part (e.g., z² instead of 2z²–x²–y²), as this is sufficient for all axial symmetries (these functions are known as “solid harmonics”). Degenerate cases are enclosed in braces.
There are mouseover effects (if your browser allows them) that should help you to identify the terms and associate them with particular quantum numbers.
```
 Symmetry of Rotations and Cartesian products
A1'  d+g+h+i     z², z⁴, x(x²-(5+2√5)y²)(x²-(5-2√5)y²), z⁶ 
A1"  i           yz(y²-(5+2√5)x²)(y²-(5-2√5)x²)(y²-(5-2√5)x²) 
A2'  R+h         R_z, y(y²-(5+2√5)x²)(y²-(5-2√5)x²)(y²-(5-2√5)x²) 
A2"  p+f+h+i     z, z³, z⁵, xz(x²-(5+2√5)y²)(x²-(5-2√5)y²) 
E1'  p+f+g+h+2i  {x, y}, {xz², yz²}, {(x²-y²)²-2x²y², xy(x²-y²)}, {xz⁴, yz⁴}, {x²(x²-3y²)²-y²(3x²-y²)², xy(x²-3y²)(3x²-y²)}, {z²((x²-y²)²-4x²y²), xyz²(x²-y²)} 
E1"  R+d+g+h+i   {R_x, R_y}, {xz, yz}, {xz³, yz³}, {z((x²-y²)²-4x²y²), xyz(x²-y²)}, {xz⁵, yz⁵} 
E2'  d+f+g+h+i   {x²-y², xy}, {x(x²-3y²), y(3x²-y²)}, {z²(x²-y²), xyz²}, {xz²(x²-3y²), yz²(3x²-y²)}, {z⁴(x²-y²), xyz⁴} 
E2"  f+g+h+i     {z(x²-y²), xyz}, {xz(x²-3y²), yz(3x²-y²)}, {z³(x²-y²), xyz³}, {xz³(x²-3y²), yz³(3x²-y²)} 
```
For cubic groups, a different set of linear combinations is used (“cubic harmonics”, though this term is also used for the basic real spherical harmonics). Only a minority of these is identical to real spherical harmonics, and therefore classification by l and m is not generally possible. Take these terms with a grain of salt, as I am not sure whether I eliminated the improper lower angular momenta correctly. For icosahedral groups, no terms are shown.
The last issue printed are notes explaining some salient features of the point group, or remarking on some issue with terminology or conventional naming of this group. The number of notes varies from group to group. Some recurrent topics are:
1. The order of the group is the number of symmetry operations in the group.
2. The order of a symmetry operation is the minimum exponent converting that symmetry operation into the unit operation (or, alternately: All symmetry operations are roots of unity of some order). The principal axis is the symmetry element giving rise to the symmetry operation with the highest order.
3. Some groups may be known by different names, although most of the alternate names will rarely occur in the more recent literature.
4. Abelian groups with a order of the principal axis greater than two have complex-charactered irreducible representations that are conventionally converted into real-charactered degenerate reducible representations (this also applies to T and T_h, but to no other cubic group). This is remarked upon because it neccessitates a small modification in the projection formula to “reduce” reducible representations into irreducible ones.
5. Some groups will display special properties due to arithmetic peculiarities of the cosine function triggered by the order of the principal axis. In particular, some have integer characters (“crystallgraphic point groups”), others have characters representable by integer numbers and square roots (“constructible polygons”).

Notational Conventions

Character tables found in various sources will usually agree on the naming of symmetry operations and irreducible representations, although they might list them in different orders. The sorting order shown in my tables is something I consider rational, but some readers might have different expectations. Give particular care to the arrangement of S_n^k operations in groups C_nh and D_nh for even n.

In the literature, noninteger characters are handled in various ways and typically represented as cosine values or by use of the complex exponential function. I consider it more convenient to show them numerically and give the exact representation in an appendix. This is really the only option for the involved radical expressions as offered for some higher groups in addition to the cosine representation.

In groups with an odd order of the principal axis, some character values appear only with positive sign, and others only with negative sign. An example is D₅, where the irrational character –1.6180 appears and is glossed as 2*cos(2π/5) in the appendix. However, in groups with an even order of the principal axis, any character value will appear with both signs; in such cases, I have chosen to show cosine representations only for the positive value. An example is D_5h (its principal axis is S₅ which has order 10): Both +1.6180 and –1.6180 are found in the table, but the only gloss given is the positive value as 2*cos(2π/10) = 2*cos(π/5). Irrational characters appear only in E races, where all character values must be of the form 2*cos(2πk/n) (n being the order of the principal axis and k ≤ n/2).

With respect to Cartesian products, authors are divided whether to show expressions relating to improper lower angular momenta in the tables. For example, one of the six possible Cartesian products of order two transforms as a scalar (x²+y² for axial and x²+y²+z² for isometric groups); it is doubtful whether such an entry is helpful in a character table, and I do not show it there.

Most sources follow the typographical convention that the subscript letters v,h,d in the names of point groups are printed in italics; I consider that ill-chosen, as italics are reserved for variables. The difference becomes clear when expressions like C_2n and C_2v are compared.

Download

The complete set of all character tables up to n=128 (in plain text format) can be downloaded as an XZ-compressed tar file. If you need other point groups, drop me a note.

Note that this is just a sparetime and fun project. There might be bugs of any magnitude; no warranty whatsoever is offered. If you find a mistake, please let me know so that I can correct it.

Algorithm

The character tables shown here were generated by a short Fortran program. Generation is done in steps, coarsely outlined by the following algorithm.

For C_n, generate a character table with a trivial A representation, an alternating B representation if n is even, and an appropriate number of E representations. The E₁ has characters equal to the double cosine of the respective rotation, 2cosφ, the E₂ has 2cos2φ etc.
For C_nv and n even, generate C_n, add two σ_v classes and double the rows for A and B (with positive or negative characters for the new classes); for n odd, add one σ_v class and double the A representation.
For C_nh and n odd, generate C_n. Then, generate new classes by multiplying the existing ones with σ_h, and double all irreducible representations, taking existing characters for the new classes and multiplying them by 1 or –1.
For C_nh and n even, do the same but use i as the generator. This procedure yields the new symmetry operations in an unintuitive order (related to the difference between mirror-reflexion and roto-inversion axes). The character tables shown on these pages have these entries resorted, which is not always done in the literature.
S_2n is identical to C_2n except the labelling of the classes. However, this is the conventional approach for S_4n only; for S_4n+2, the character table is built from C_2n+1 by adding a center of inversion similar to C_2nh.
For D_n, generate C_n. Then, add C₂ rotations similar to the C_nv case.
For D_nh, generate D_n. Then, proceed as in C_nh.
For D_nd and odd n, generate D_n, then double the table by adding an additional center of inversion (as in C_nh).
For D_nd and even n, the conventional form of the character table is arrived at by starting with S_n, then adding one C₂^′ and one σ_d class, similar to C_nv. This special treatment is necessary because while D_2n has distinct C₂^′ and C₂^″ classes, these collapse into one single C₂^′ in the D_2nd group.
Similar procedures can be used to build T_d, T_h and O from T, O_h from O and I_h from I.

Labelling and sorting needs some care, because conventions differ for different point groups.

The transformation of rotations and Cartesian products can be arrived at easily for C_n and S_2n: Both components of a ±m pair (e.g., d_±2 aka x²–y² and xy) will always belong to the same symmetry race; moreover, the characters for the combined representation of one ±m pair are just 2cos|m|φ (add a sign (–1)^l–|m| for S_2n, where l is the order of the product or the angular momentum quantum number of the corresponding spherical harmonics). Since cos(0)=1, the m=0 components always transform trivially in C_n, while in S_2n this is true only if l is even.

When the groups are subsequently augmented with additional symmetry operations, I chose to keep track of the various steps of irrep doublings, and move the basis functions to the positive or negative offspring row, depending on the behaviour of that basis function under the newly introduced symmetry element (I used a big table for that). This approach might appear unelegant, but has the advantage that neither complex arithmetics nor case distictions between one- and two-dimensional symmetry races are needed.

Real radicals

All irrational character values have the form 2*cos(2*m*π/n), where n is the order of the principal axis and m<n. For some point groups (n=3,4,5,6,8,10,12,15,16,17,20,24,30,32,34,40,48,51,64,68,80,85,96,102,120,128,136,160,170,192,240,255,256,257,…), this can also be written with real radicals.

I have managed to derive algebraic expressions for almost all cases where this is possible (n=85 and its multiples are still missing). Possibly, some of these could be further simplified, but I am not motivated. For your pleasure, here is the full list in a format most programming languages (or bc) can directly understand. If you want to see them in conventional form with graphic square roots, follow the link on the cosine term (don’t expect typographic excellence, it’s HTML and not TEX).

  0.024543076571 = 2*cos(127*pi/256) = sqrt(2-sqrt(2+sqrt(2+sqrt(2+sqrt(2+sqrt(2+sqrt(2)))))))
  0.032723463253 = 2*cos(95*pi/192) = sqrt(2-sqrt(2+sqrt(2+sqrt(2+sqrt(2+sqrt(3))))))
  0.049082457046 = 2*cos(63*pi/128) = sqrt(2-sqrt(2+sqrt(2+sqrt(2+sqrt(2+sqrt(2))))))
  0.061590117112 = 2*cos(25*pi/51) = (-1-sqrt(17)-sqrt(34+2*sqrt(17))-2*sqrt(17-3*sqrt(17)+sqrt(34+2*sqrt(17))-2*sqrt(34-2*sqrt(17))))/16 + sqrt(6)*sqrt(17+sqrt(17)-sqrt(34+2*sqrt(17))-2*sqrt(17-3*sqrt(17)-sqrt(34+2*sqrt(17))+2*sqrt(34-2*sqrt(17))))/8
  0.065438165643 = 2*cos(47*pi/96) = sqrt(2-sqrt(2+sqrt(2+sqrt(2+sqrt(3)))))
  0.073614445883 = 2*cos(125*pi/256) = sqrt(2-sqrt(2+sqrt(2+sqrt(2+sqrt(2+sqrt(2-sqrt(2)))))))
  0.078519631518 = 2*cos(39*pi/80) = sqrt(4-sqrt(8+sqrt(2)+sqrt(10)+2*sqrt(5-sqrt(5))))/sqrt(2)
  0.092366917291 = 2*cos(33*pi/68) = sqrt(8-sqrt(2)*sqrt(17-sqrt(17)+sqrt(34-2*sqrt(17))+2*sqrt(17+3*sqrt(17)+sqrt(34-2*sqrt(17))+2*sqrt(34+2*sqrt(17)))))/2
  0.098135348655 = 2*cos(31*pi/64) = sqrt(2-sqrt(2+sqrt(2+sqrt(2+sqrt(2)))))
  0.104671912486 = 2*cos(29*pi/60) = (sqrt(10)-sqrt(2)-sqrt(6)+sqrt(30)+2*(1-sqrt(3))*(sqrt(5+sqrt(5))))/8
  0.122641472604 = 2*cos(123*pi/256) = sqrt(2-sqrt(2+sqrt(2+sqrt(2+sqrt(2-sqrt(2-sqrt(2)))))))
  0.130806258460 = 2*cos(23*pi/48) = sqrt(2-sqrt(2+sqrt(2+sqrt(3))))
  0.147129127199 = 2*cos(61*pi/128) = sqrt(2-sqrt(2+sqrt(2+sqrt(2+sqrt(2-sqrt(2))))))
  0.156918191455 = 2*cos(19*pi/40) = sqrt(8-sqrt(2)-sqrt(10)-2*sqrt(5-sqrt(5)))/2
  0.163442148267 = 2*cos(91*pi/192) = sqrt(2-sqrt(2+sqrt(2+sqrt(2+sqrt(2-sqrt(3))))))
  0.171594624689 = 2*cos(121*pi/256) = sqrt(2-sqrt(2+sqrt(2+sqrt(2+sqrt(2-sqrt(2+sqrt(2)))))))
  0.184536718926 = 2*cos(8*pi/17) = (-1+sqrt(17)+sqrt(34-2*sqrt(17))-2*sqrt(17+3*sqrt(17)-sqrt(34-2*sqrt(17))-2*sqrt(34+2*sqrt(17))))/8
  0.196034280659 = 2*cos(15*pi/32) = sqrt(2-sqrt(2+sqrt(2+sqrt(2))))
  0.209056926535 = 2*cos(7*pi/15) = (sqrt(30-6*sqrt(5))-sqrt(5)-1)/4
  0.220444414588 = 2*cos(119*pi/256) = sqrt(2-sqrt(2+sqrt(2+sqrt(2-sqrt(2-sqrt(2+sqrt(2)))))))
  0.228573929934 = 2*cos(89*pi/192) = sqrt(2-sqrt(2+sqrt(2+sqrt(2-sqrt(2-sqrt(3))))))
  0.235074794915 = 2*cos(37*pi/80) = sqrt(4-sqrt(8-sqrt(2)+sqrt(10)+2*sqrt(5+sqrt(5))))/sqrt(2)
  0.244821350398 = 2*cos(59*pi/128) = sqrt(2-sqrt(2+sqrt(2+sqrt(2-sqrt(2-sqrt(2))))))
  0.261052384440 = 2*cos(11*pi/24) = sqrt(2-sqrt(2+sqrt(3)))
  0.269161417014 = 2*cos(117*pi/256) = sqrt(2-sqrt(2+sqrt(2+sqrt(2-sqrt(2-sqrt(2-sqrt(2)))))))
  0.276312709904 = 2*cos(31*pi/68) = sqrt(8-sqrt(2)*sqrt(17+sqrt(17)+sqrt(34+2*sqrt(17))+2*sqrt(17-3*sqrt(17)+sqrt(34+2*sqrt(17))-2*sqrt(34-2*sqrt(17)))))/2
  0.293460948911 = 2*cos(29*pi/64) = sqrt(2-sqrt(2+sqrt(2+sqrt(2-sqrt(2)))))
  0.306783309757 = 2*cos(23*pi/51) = (-1+sqrt(17)+sqrt(34-2*sqrt(17))+2*sqrt(17+3*sqrt(17)-sqrt(34-2*sqrt(17))-2*sqrt(34+2*sqrt(17))))/16 - sqrt(6)*sqrt(17-sqrt(17)+sqrt(34-2*sqrt(17))-2*sqrt(17+3*sqrt(17)+sqrt(34-2*sqrt(17))+2*sqrt(34+2*sqrt(17))))/8
  0.312868930080 = 2*cos(9*pi/20) = (sqrt(2)+sqrt(10)-2*(sqrt(5-sqrt(5))))/4
  0.317716286668 = 2*cos(115*pi/256) = sqrt(2-sqrt(2+sqrt(2+sqrt(2-sqrt(2+sqrt(2-sqrt(2)))))))
  0.325790946789 = 2*cos(43*pi/96) = sqrt(2-sqrt(2+sqrt(2+sqrt(2-sqrt(3)))))
  0.341923777520 = 2*cos(57*pi/128) = sqrt(2-sqrt(2+sqrt(2+sqrt(2-sqrt(2+sqrt(2))))))
  0.358033722553 = 2*cos(85*pi/192) = sqrt(2-sqrt(2+sqrt(2+sqrt(2-sqrt(2+sqrt(3))))))
  0.366079775910 = 2*cos(113*pi/256) = sqrt(2-sqrt(2+sqrt(2+sqrt(2-sqrt(2+sqrt(2+sqrt(2)))))))
  0.367499035633 = 2*cos(15*pi/34) = sqrt(2)*sqrt(17-sqrt(17)-sqrt(34-2*sqrt(17))-2*sqrt(17+3*sqrt(17)-sqrt(34-2*sqrt(17))-2*sqrt(34+2*sqrt(17))))/4
  0.390180644032 = 2*cos(7*pi/16) = sqrt(2-sqrt(2+sqrt(2)))
  0.414222752384 = 2*cos(111*pi/256) = sqrt(2-sqrt(2+sqrt(2-sqrt(2-sqrt(2+sqrt(2+sqrt(2)))))))
  0.415823381635 = 2*cos(13*pi/30) = (sqrt(3)-sqrt(15)+sqrt(10+2*sqrt(5)))/4
  0.422223104718 = 2*cos(83*pi/192) = sqrt(2-sqrt(2+sqrt(2-sqrt(2-sqrt(2+sqrt(3))))))
  0.427866166413 = 2*cos(22*pi/51) = (1-sqrt(17)+sqrt(34-2*sqrt(17))-2*sqrt(17+3*sqrt(17)+sqrt(34-2*sqrt(17))+2*sqrt(34+2*sqrt(17))))/16 + sqrt(6)*sqrt(17-sqrt(17)-sqrt(34-2*sqrt(17))+2*sqrt(17+3*sqrt(17)-sqrt(34-2*sqrt(17))-2*sqrt(34+2*sqrt(17))))/8
  0.438202480314 = 2*cos(55*pi/128) = sqrt(2-sqrt(2+sqrt(2-sqrt(2-sqrt(2+sqrt(2))))))
  0.454152526069 = 2*cos(41*pi/96) = sqrt(2-sqrt(2+sqrt(2-sqrt(2-sqrt(3)))))
  0.457901099900 = 2*cos(29*pi/68) = sqrt(8-sqrt(2)*sqrt(17+sqrt(17)+sqrt(34+2*sqrt(17))-2*sqrt(17-3*sqrt(17)+sqrt(34+2*sqrt(17))-2*sqrt(34-2*sqrt(17)))))/2
  0.462116216561 = 2*cos(109*pi/256) = sqrt(2-sqrt(2+sqrt(2-sqrt(2-sqrt(2+sqrt(2-sqrt(2)))))))
  0.466890727712 = 2*cos(17*pi/40) = sqrt(8+sqrt(2)-sqrt(10)-2*sqrt(5+sqrt(5)))/2
  0.485960359806 = 2*cos(27*pi/64) = sqrt(2-sqrt(2+sqrt(2-sqrt(2-sqrt(2)))))
  0.509731319209 = 2*cos(107*pi/256) = sqrt(2-sqrt(2+sqrt(2-sqrt(2-sqrt(2-sqrt(2-sqrt(2)))))))
  0.517638090205 = 2*cos(5*pi/12) = sqrt(2-sqrt(3))
  0.533425514950 = 2*cos(53*pi/128) = sqrt(2-sqrt(2+sqrt(2-sqrt(2-sqrt(2-sqrt(2))))))
  0.542880899730 = 2*cos(33*pi/80) = sqrt(4-sqrt(8+sqrt(2)-sqrt(10)+2*sqrt(5+sqrt(5))))/sqrt(2)
  0.547325980144 = 2*cos(7*pi/17) = (1+sqrt(17)-sqrt(34+2*sqrt(17))+2*sqrt(17-3*sqrt(17)-sqrt(34+2*sqrt(17))+2*sqrt(34-2*sqrt(17))))/8
  0.549177236370 = 2*cos(79*pi/192) = sqrt(2-sqrt(2+sqrt(2-sqrt(2-sqrt(2-sqrt(3))))))
  0.557039378770 = 2*cos(105*pi/256) = sqrt(2-sqrt(2+sqrt(2-sqrt(2-sqrt(2-sqrt(2+sqrt(2)))))))
  0.580569354509 = 2*cos(13*pi/32) = sqrt(2-sqrt(2+sqrt(2-sqrt(2))))
  0.604011898638 = 2*cos(103*pi/256) = sqrt(2-sqrt(2+sqrt(2-sqrt(2+sqrt(2-sqrt(2+sqrt(2)))))))
  0.611806040193 = 2*cos(77*pi/192) = sqrt(2-sqrt(2+sqrt(2-sqrt(2+sqrt(2-sqrt(3))))))
  0.618033988750 = 2*cos(2*pi/5) = (sqrt(5)-1)/2
  0.627363480798 = 2*cos(51*pi/128) = sqrt(2-sqrt(2+sqrt(2-sqrt(2+sqrt(2-sqrt(2))))))
  0.635582839164 = 2*cos(27*pi/68) = sqrt(8-sqrt(2)*sqrt(17+sqrt(17)-sqrt(34+2*sqrt(17))+2*sqrt(17-3*sqrt(17)-sqrt(34+2*sqrt(17))+2*sqrt(34-2*sqrt(17)))))/2
  0.642878930606 = 2*cos(19*pi/48) = sqrt(2-sqrt(2+sqrt(2-sqrt(3))))
  0.650620584324 = 2*cos(101*pi/256) = sqrt(2-sqrt(2+sqrt(2-sqrt(2+sqrt(2-sqrt(2-sqrt(2)))))))
  0.664709598959 = 2*cos(20*pi/51) = (1-sqrt(17)+sqrt(34-2*sqrt(17))+2*sqrt(17+3*sqrt(17)+sqrt(34-2*sqrt(17))+2*sqrt(34+2*sqrt(17))))/16 - sqrt(6)*sqrt(17-sqrt(17)-sqrt(34-2*sqrt(17))-2*sqrt(17+3*sqrt(17)-sqrt(34-2*sqrt(17))-2*sqrt(34+2*sqrt(17))))/8
  0.673779706784 = 2*cos(25*pi/64) = sqrt(2-sqrt(2+sqrt(2-sqrt(2+sqrt(2)))))
  0.692234114155 = 2*cos(31*pi/80) = sqrt(4-sqrt(8+sqrt(2)+sqrt(10)-2*sqrt(5-sqrt(5))))/sqrt(2)
  0.696837360499 = 2*cos(99*pi/256) = sqrt(2-sqrt(2+sqrt(2-sqrt(2+sqrt(2+sqrt(2-sqrt(2)))))))
  0.704500095842 = 2*cos(37*pi/96) = sqrt(2-sqrt(2+sqrt(2-sqrt(2+sqrt(3)))))
  0.716735899090 = 2*cos(23*pi/60) = (sqrt(2)-sqrt(6)+sqrt(10)-sqrt(30)+2*(1+sqrt(3))*(sqrt(5-sqrt(5))))/8
  0.719790073070 = 2*cos(49*pi/128) = sqrt(2-sqrt(2+sqrt(2-sqrt(2+sqrt(2+sqrt(2))))))
  0.722483332374 = 2*cos(13*pi/34) = sqrt(2)*sqrt(17-sqrt(17)+sqrt(34-2*sqrt(17))-2*sqrt(17+3*sqrt(17)+sqrt(34-2*sqrt(17))+2*sqrt(34+2*sqrt(17))))/4
  0.735031873189 = 2*cos(73*pi/192) = sqrt(2-sqrt(2+sqrt(2-sqrt(2+sqrt(2+sqrt(3))))))
  0.742634387904 = 2*cos(97*pi/256) = sqrt(2-sqrt(2+sqrt(2-sqrt(2+sqrt(2+sqrt(2+sqrt(2)))))))
  0.765366864730 = 2*cos(3*pi/8) = sqrt(2-sqrt(2))
  0.779571746585 = 2*cos(19*pi/51) = (-1-sqrt(17)-sqrt(34+2*sqrt(17))+2*sqrt(17-3*sqrt(17)+sqrt(34+2*sqrt(17))-2*sqrt(34-2*sqrt(17))))/16 + sqrt(6)*sqrt(17+sqrt(17)-sqrt(34+2*sqrt(17))+2*sqrt(17-3*sqrt(17)-sqrt(34+2*sqrt(17))+2*sqrt(34-2*sqrt(17))))/8
  0.787984080122 = 2*cos(95*pi/256) = sqrt(2-sqrt(2-sqrt(2-sqrt(2+sqrt(2+sqrt(2+sqrt(2)))))))
  0.795496949054 = 2*cos(71*pi/192) = sqrt(2-sqrt(2-sqrt(2-sqrt(2+sqrt(2+sqrt(3))))))
  0.807842009744 = 2*cos(25*pi/68) = sqrt(8-sqrt(2)*sqrt(17-sqrt(17)-sqrt(34-2*sqrt(17))+2*sqrt(17+3*sqrt(17)-sqrt(34-2*sqrt(17))-2*sqrt(34+2*sqrt(17)))))/2
  0.810482628010 = 2*cos(47*pi/128) = sqrt(2-sqrt(2-sqrt(2-sqrt(2+sqrt(2+sqrt(2))))))
  0.813473286151 = 2*cos(11*pi/30) = (sqrt(15)+sqrt(3)-sqrt(10-2*sqrt(5)))/4
  0.825414059609 = 2*cos(35*pi/96) = sqrt(2-sqrt(2-sqrt(2-sqrt(2+sqrt(3)))))
  0.832859120195 = 2*cos(93*pi/256) = sqrt(2-sqrt(2-sqrt(2-sqrt(2+sqrt(2+sqrt(2-sqrt(2)))))))
  0.837319475075 = 2*cos(29*pi/80) = sqrt(4-sqrt(8-sqrt(2)-sqrt(10)+2*sqrt(5-sqrt(5))))/sqrt(2)
  0.855110186860 = 2*cos(23*pi/64) = sqrt(2-sqrt(2-sqrt(2-sqrt(2+sqrt(2)))))
  0.877232477077 = 2*cos(91*pi/256) = sqrt(2-sqrt(2-sqrt(2-sqrt(2+sqrt(2-sqrt(2-sqrt(2)))))))
  0.884577380438 = 2*cos(17*pi/48) = sqrt(2-sqrt(2-sqrt(2-sqrt(3))))
  0.891476711553 = 2*cos(6*pi/17) = (-1-sqrt(17)+sqrt(34+2*sqrt(17))+2*sqrt(17-3*sqrt(17)-sqrt(34+2*sqrt(17))+2*sqrt(34-2*sqrt(17))))/8
  0.899222659309 = 2*cos(45*pi/128) = sqrt(2-sqrt(2-sqrt(2-sqrt(2+sqrt(2-sqrt(2))))))
  0.907980999479 = 2*cos(7*pi/20) = (sqrt(2)-sqrt(10)+2*sqrt(5+sqrt(5)))/4
  0.913807751261 = 2*cos(67*pi/192) = sqrt(2-sqrt(2-sqrt(2-sqrt(2+sqrt(2-sqrt(3))))))
  0.921077421916 = 2*cos(89*pi/256) = sqrt(2-sqrt(2-sqrt(2-sqrt(2+sqrt(2-sqrt(2+sqrt(2)))))))
  0.942793473652 = 2*cos(11*pi/32) = sqrt(2-sqrt(2-sqrt(2-sqrt(2))))
  0.964367544158 = 2*cos(87*pi/256) = sqrt(2-sqrt(2-sqrt(2-sqrt(2-sqrt(2-sqrt(2+sqrt(2)))))))
  0.971526787433 = 2*cos(65*pi/192) = sqrt(2-sqrt(2-sqrt(2-sqrt(2-sqrt(2-sqrt(3))))))
  0.973208957134 = 2*cos(23*pi/68) = sqrt(8-sqrt(2)*sqrt(17+sqrt(17)-sqrt(34+2*sqrt(17))-2*sqrt(17-3*sqrt(17)-sqrt(34+2*sqrt(17))+2*sqrt(34-2*sqrt(17)))))/2
  0.977242482994 = 2*cos(27*pi/80) = sqrt(4-sqrt(8-sqrt(2)+sqrt(10)-2*sqrt(5+sqrt(5))))/sqrt(2)
  0.985796384459 = 2*cos(43*pi/128) = sqrt(2-sqrt(2-sqrt(2-sqrt(2-sqrt(2-sqrt(2))))))
  1.007076767451 = 2*cos(85*pi/256) = sqrt(2-sqrt(2-sqrt(2-sqrt(2-sqrt(2-sqrt(2-sqrt(2)))))))
  1.028205488386 = 2*cos(21*pi/64) = sqrt(2-sqrt(2-sqrt(2-sqrt(2-sqrt(2)))))
  1.044997129432 = 2*cos(13*pi/40) = sqrt(8-sqrt(2)+sqrt(10)-2*sqrt(5+sqrt(5)))/2
  1.049179365357 = 2*cos(83*pi/256) = sqrt(2-sqrt(2-sqrt(2-sqrt(2-sqrt(2+sqrt(2-sqrt(2)))))))
  1.052864325755 = 2*cos(11*pi/34) = sqrt(2)*sqrt(17+sqrt(17)-sqrt(34+2*sqrt(17))-2*sqrt(17-3*sqrt(17)-sqrt(34+2*sqrt(17))+2*sqrt(34-2*sqrt(17))))/4
  1.056135701301 = 2*cos(31*pi/96) = sqrt(2-sqrt(2-sqrt(2-sqrt(2-sqrt(3)))))
  1.069995239774 = 2*cos(41*pi/128) = sqrt(2-sqrt(2-sqrt(2-sqrt(2-sqrt(2+sqrt(2))))))
  1.083783161149 = 2*cos(61*pi/192) = sqrt(2-sqrt(2-sqrt(2-sqrt(2-sqrt(2+sqrt(3))))))
  1.089278070030 = 2*cos(19*pi/60) = (sqrt(10)-sqrt(2)-sqrt(6)+sqrt(30)-2*(1-sqrt(3))*(sqrt(5+sqrt(5))))/8
  1.090649976844 = 2*cos(81*pi/256) = sqrt(2-sqrt(2-sqrt(2-sqrt(2-sqrt(2+sqrt(2+sqrt(2)))))))
  1.104729945921 = 2*cos(16*pi/51) = (1+sqrt(17)-sqrt(34+2*sqrt(17))-2*sqrt(17-3*sqrt(17)-sqrt(34+2*sqrt(17))+2*sqrt(34-2*sqrt(17))))/16 + sqrt(6)*sqrt(17+sqrt(17)+sqrt(34+2*sqrt(17))-2*sqrt(17-3*sqrt(17)+sqrt(34+2*sqrt(17))-2*sqrt(34-2*sqrt(17))))/8
  1.111140466039 = 2*cos(5*pi/16) = sqrt(2-sqrt(2-sqrt(2)))
  1.130272828845 = 2*cos(21*pi/68) = sqrt(8-sqrt(2)*sqrt(17-sqrt(17)+sqrt(34-2*sqrt(17))-2*sqrt(17+3*sqrt(17)+sqrt(34-2*sqrt(17))+2*sqrt(34+2*sqrt(17)))))/2
  1.131463621567 = 2*cos(79*pi/256) = sqrt(2-sqrt(2-sqrt(2+sqrt(2-sqrt(2+sqrt(2+sqrt(2)))))))
  1.138200291758 = 2*cos(59*pi/192) = sqrt(2-sqrt(2-sqrt(2+sqrt(2-sqrt(2+sqrt(3))))))
  1.151616382836 = 2*cos(39*pi/128) = sqrt(2-sqrt(2-sqrt(2+sqrt(2-sqrt(2+sqrt(2))))))
  1.164955393736 = 2*cos(29*pi/96) = sqrt(2-sqrt(2-sqrt(2+sqrt(2-sqrt(3)))))
  1.171595714913 = 2*cos(77*pi/256) = sqrt(2-sqrt(2-sqrt(2+sqrt(2-sqrt(2+sqrt(2-sqrt(2)))))))
  1.175570504585 = 2*cos(3*pi/10) = (sqrt(10-2*sqrt(5)))/2
  1.191398608985 = 2*cos(19*pi/64) = sqrt(2-sqrt(2-sqrt(2+sqrt(2-sqrt(2)))))
  1.205269272758 = 2*cos(5*pi/17) = (1+sqrt(17)+sqrt(34+2*sqrt(17))-2*sqrt(17-3*sqrt(17)+sqrt(34+2*sqrt(17))-2*sqrt(34-2*sqrt(17))))/8
  1.211022082809 = 2*cos(75*pi/256) = sqrt(2-sqrt(2-sqrt(2+sqrt(2-sqrt(2-sqrt(2-sqrt(2)))))))
  1.217522858017 = 2*cos(7*pi/24) = sqrt(2-sqrt(2-sqrt(3)))
  1.230463181161 = 2*cos(37*pi/128) = sqrt(2-sqrt(2-sqrt(2+sqrt(2-sqrt(2-sqrt(2))))))
  1.238187898620 = 2*cos(23*pi/80) = sqrt(4-sqrt(8+sqrt(2)-sqrt(10)-2*sqrt(5+sqrt(5))))/sqrt(2)
  1.243321146740 = 2*cos(55*pi/192) = sqrt(2-sqrt(2-sqrt(2+sqrt(2-sqrt(2-sqrt(3))))))
  1.249718976285 = 2*cos(73*pi/256) = sqrt(2-sqrt(2-sqrt(2+sqrt(2-sqrt(2-sqrt(2+sqrt(2)))))))
  1.258640782100 = 2*cos(17*pi/60) = (sqrt(2)+sqrt(6)+sqrt(10)+sqrt(30)+2*(1-sqrt(3))*(sqrt(5-sqrt(5))))/8
  1.268786568327 = 2*cos(9*pi/32) = sqrt(2-sqrt(2-sqrt(2+sqrt(2))))
  1.277693611304 = 2*cos(19*pi/68) = sqrt(8-sqrt(2)*sqrt(17-sqrt(17)-sqrt(34-2*sqrt(17))-2*sqrt(17+3*sqrt(17)-sqrt(34-2*sqrt(17))-2*sqrt(34+2*sqrt(17)))))/2
  1.287663085779 = 2*cos(71*pi/256) = sqrt(2-sqrt(2-sqrt(2+sqrt(2+sqrt(2-sqrt(2+sqrt(2)))))))
  1.293912305070 = 2*cos(53*pi/192) = sqrt(2-sqrt(2-sqrt(2+sqrt(2+sqrt(2-sqrt(3))))))
  1.298896096660 = 2*cos(11*pi/40) = sqrt(8-sqrt(2)-sqrt(10)+2*sqrt(5-sqrt(5)))/2
  1.301236600408 = 2*cos(14*pi/51) = (1-sqrt(17)+sqrt(34-2*sqrt(17))+2*sqrt(17+3*sqrt(17)+sqrt(34-2*sqrt(17))+2*sqrt(34+2*sqrt(17))))/16 + sqrt(6)*sqrt(17-sqrt(17)-sqrt(34-2*sqrt(17))-2*sqrt(17+3*sqrt(17)-sqrt(34-2*sqrt(17))-2*sqrt(34+2*sqrt(17))))/8
  1.306345685907 = 2*cos(35*pi/128) = sqrt(2-sqrt(2-sqrt(2+sqrt(2+sqrt(2-sqrt(2))))))
  1.318691630200 = 2*cos(13*pi/48) = sqrt(2-sqrt(2-sqrt(2+sqrt(3))))
  1.324831555180 = 2*cos(69*pi/256) = sqrt(2-sqrt(2-sqrt(2+sqrt(2+sqrt(2-sqrt(2-sqrt(2)))))))
  1.338261212718 = 2*cos(4*pi/15) = (1-sqrt(5)+sqrt(30+6*sqrt(5)))/4
  1.343117909694 = 2*cos(17*pi/64) = sqrt(2-sqrt(2-sqrt(2+sqrt(2+sqrt(2)))))
  1.347391287293 = 2*cos(9*pi/34) = sqrt(2)*sqrt(17-sqrt(17)-sqrt(34-2*sqrt(17))+2*sqrt(17+3*sqrt(17)-sqrt(34-2*sqrt(17))-2*sqrt(34+2*sqrt(17))))/4
  1.357601491066 = 2*cos(21*pi/80) = sqrt(4-sqrt(8-sqrt(2)-sqrt(10)-2*sqrt(5-sqrt(5))))/sqrt(2)
  1.361201995591 = 2*cos(67*pi/256) = sqrt(2-sqrt(2-sqrt(2+sqrt(2+sqrt(2+sqrt(2-sqrt(2)))))))
  1.367184604046 = 2*cos(25*pi/96) = sqrt(2-sqrt(2-sqrt(2+sqrt(2+sqrt(3)))))
  1.379081089474 = 2*cos(33*pi/128) = sqrt(2-sqrt(2-sqrt(2+sqrt(2+sqrt(2+sqrt(2))))))
  1.390885270019 = 2*cos(49*pi/192) = sqrt(2-sqrt(2-sqrt(2+sqrt(2+sqrt(2+sqrt(3))))))
  1.392267891926 = 2*cos(13*pi/51) = (-1-sqrt(17)+sqrt(34+2*sqrt(17))-2*sqrt(17-3*sqrt(17)-sqrt(34+2*sqrt(17))+2*sqrt(34-2*sqrt(17))))/16 + sqrt(6)*sqrt(17+sqrt(17)+sqrt(34+2*sqrt(17))+2*sqrt(17-3*sqrt(17)+sqrt(34+2*sqrt(17))-2*sqrt(34-2*sqrt(17))))/8
  1.396752498818 = 2*cos(65*pi/256) = sqrt(2-sqrt(2-sqrt(2+sqrt(2+sqrt(2+sqrt(2+sqrt(2)))))))
  1.414213562373 = 2*cos(pi/4) = sqrt(2)
  1.431461650568 = 2*cos(63*pi/256) = sqrt(2+sqrt(2-sqrt(2+sqrt(2+sqrt(2+sqrt(2+sqrt(2)))))))
  1.437163235559 = 2*cos(47*pi/192) = sqrt(2+sqrt(2-sqrt(2+sqrt(2+sqrt(2+sqrt(3))))))
  1.448494165903 = 2*cos(31*pi/128) = sqrt(2+sqrt(2-sqrt(2+sqrt(2+sqrt(2+sqrt(2))))))
  1.459728145396 = 2*cos(23*pi/96) = sqrt(2+sqrt(2-sqrt(2+sqrt(2+sqrt(3)))))
  1.465308543345 = 2*cos(61*pi/256) = sqrt(2+sqrt(2-sqrt(2+sqrt(2+sqrt(2+sqrt(2-sqrt(2)))))))
  1.468645018871 = 2*cos(19*pi/80) = sqrt(4+sqrt(8-sqrt(2)-sqrt(10)-2*sqrt(5-sqrt(5))))/sqrt(2)
  1.478017834441 = 2*cos(4*pi/17) = (-1+sqrt(17)-sqrt(34-2*sqrt(17))+2*sqrt(17+3*sqrt(17)+sqrt(34-2*sqrt(17))+2*sqrt(34+2*sqrt(17))))/8
  1.481902250710 = 2*cos(15*pi/64) = sqrt(2+sqrt(2-sqrt(2+sqrt(2+sqrt(2)))))
  1.486289650955 = 2*cos(7*pi/30) = (sqrt(15)-sqrt(3)+sqrt(10+2*sqrt(5)))/4
  1.498272789047 = 2*cos(59*pi/256) = sqrt(2+sqrt(2-sqrt(2+sqrt(2+sqrt(2-sqrt(2-sqrt(2)))))))
  1.503679614958 = 2*cos(11*pi/48) = sqrt(2+sqrt(2-sqrt(2+sqrt(3))))
  1.514417693013 = 2*cos(29*pi/128) = sqrt(2+sqrt(2-sqrt(2+sqrt(2+sqrt(2-sqrt(2))))))
  1.520811931200 = 2*cos(9*pi/40) = sqrt(8+sqrt(2)+sqrt(10)-2*sqrt(5-sqrt(5)))/2
  1.525054407813 = 2*cos(43*pi/192) = sqrt(2+sqrt(2-sqrt(2+sqrt(2+sqrt(2-sqrt(3))))))
  1.530334531245 = 2*cos(57*pi/256) = sqrt(2+sqrt(2-sqrt(2+sqrt(2+sqrt(2-sqrt(2+sqrt(2)))))))
  1.538667941966 = 2*cos(15*pi/68) = sqrt(8+sqrt(2)*sqrt(17-sqrt(17)-sqrt(34-2*sqrt(17))-2*sqrt(17+3*sqrt(17)-sqrt(34-2*sqrt(17))-2*sqrt(34+2*sqrt(17)))))/2
  1.546020906725 = 2*cos(7*pi/32) = sqrt(2+sqrt(2-sqrt(2+sqrt(2))))
  1.554291922914 = 2*cos(13*pi/60) = (sqrt(6)-sqrt(2)-sqrt(10)+sqrt(30)+2*(1+sqrt(3))*(sqrt(5-sqrt(5))))/8
  1.558161149051 = 2*cos(11*pi/51) = (-1+sqrt(17)+sqrt(34-2*sqrt(17))+2*sqrt(17+3*sqrt(17)-sqrt(34-2*sqrt(17))-2*sqrt(34+2*sqrt(17))))/16 + sqrt(6)*sqrt(17-sqrt(17)+sqrt(34-2*sqrt(17))-2*sqrt(17+3*sqrt(17)+sqrt(34-2*sqrt(17))+2*sqrt(34+2*sqrt(17))))/8
  1.561474457144 = 2*cos(55*pi/256) = sqrt(2+sqrt(2-sqrt(2+sqrt(2-sqrt(2-sqrt(2+sqrt(2)))))))
  1.566573498457 = 2*cos(41*pi/192) = sqrt(2+sqrt(2-sqrt(2+sqrt(2-sqrt(2-sqrt(3))))))
  1.570633861761 = 2*cos(17*pi/80) = sqrt(4+sqrt(8+sqrt(2)-sqrt(10)-2*sqrt(5+sqrt(5))))/sqrt(2)
  1.576692855253 = 2*cos(27*pi/128) = sqrt(2+sqrt(2-sqrt(2+sqrt(2-sqrt(2-sqrt(2))))))
  1.586706680582 = 2*cos(5*pi/24) = sqrt(2+sqrt(2-sqrt(3)))
  1.591673809218 = 2*cos(53*pi/256) = sqrt(2+sqrt(2-sqrt(2+sqrt(2-sqrt(2-sqrt(2-sqrt(2)))))))
  1.596034454560 = 2*cos(7*pi/34) = sqrt(2)*sqrt(17+sqrt(17)-sqrt(34+2*sqrt(17))+2*sqrt(17-3*sqrt(17)-sqrt(34+2*sqrt(17))+2*sqrt(34-2*sqrt(17))))/4
  1.606415062961 = 2*cos(13*pi/64) = sqrt(2+sqrt(2-sqrt(2+sqrt(2-sqrt(2)))))
  1.618033988750 = 2*cos(pi/5) = (1+sqrt(5))/2
  1.620914396505 = 2*cos(51*pi/256) = sqrt(2+sqrt(2-sqrt(2+sqrt(2-sqrt(2+sqrt(2-sqrt(2)))))))
  1.625693369183 = 2*cos(19*pi/96) = sqrt(2+sqrt(2-sqrt(2+sqrt(2-sqrt(3)))))
  1.632393824712 = 2*cos(10*pi/51) = (1-sqrt(17)-sqrt(34-2*sqrt(17))+2*sqrt(17+3*sqrt(17)-sqrt(34-2*sqrt(17))-2*sqrt(34+2*sqrt(17))))/16 + sqrt(6)*sqrt(17-sqrt(17)+sqrt(34-2*sqrt(17))+2*sqrt(17+3*sqrt(17)+sqrt(34-2*sqrt(17))+2*sqrt(34+2*sqrt(17))))/8
  1.635169626303 = 2*cos(25*pi/128) = sqrt(2+sqrt(2-sqrt(2+sqrt(2-sqrt(2+sqrt(2))))))
  1.644536437980 = 2*cos(37*pi/192) = sqrt(2+sqrt(2-sqrt(2+sqrt(2-sqrt(2+sqrt(3))))))
  1.649178605570 = 2*cos(49*pi/256) = sqrt(2+sqrt(2-sqrt(2+sqrt(2-sqrt(2+sqrt(2+sqrt(2)))))))
  1.649994949197 = 2*cos(13*pi/68) = sqrt(8+sqrt(2)*sqrt(17-sqrt(17)+sqrt(34-2*sqrt(17))-2*sqrt(17+3*sqrt(17)+sqrt(34-2*sqrt(17))+2*sqrt(34+2*sqrt(17)))))/2
  1.662939224605 = 2*cos(3*pi/16) = sqrt(2+sqrt(2-sqrt(2)))
  1.676449411110 = 2*cos(47*pi/256) = sqrt(2+sqrt(2-sqrt(2-sqrt(2-sqrt(2+sqrt(2+sqrt(2)))))))
  1.677341135891 = 2*cos(11*pi/60) = (sqrt(6)-sqrt(2)+sqrt(10)-sqrt(30)+2*(1+sqrt(3))*(sqrt(5+sqrt(5))))/8
  1.680896802189 = 2*cos(35*pi/192) = sqrt(2+sqrt(2-sqrt(2-sqrt(2-sqrt(2+sqrt(3))))))
  1.689707130499 = 2*cos(23*pi/128) = sqrt(2+sqrt(2-sqrt(2-sqrt(2-sqrt(2+sqrt(2))))))
  1.698404363053 = 2*cos(17*pi/96) = sqrt(2+sqrt(2-sqrt(2-sqrt(2-sqrt(3)))))
  1.700434271459 = 2*cos(3*pi/17) = (1+sqrt(17)+sqrt(34+2*sqrt(17))+2*sqrt(17-3*sqrt(17)+sqrt(34+2*sqrt(17))-2*sqrt(34-2*sqrt(17))))/8
  1.702710386210 = 2*cos(45*pi/256) = sqrt(2+sqrt(2-sqrt(2-sqrt(2-sqrt(2+sqrt(2-sqrt(2)))))))
  1.705280328708 = 2*cos(7*pi/40) = sqrt(8+sqrt(2)-sqrt(10)+2*sqrt(5+sqrt(5)))/2
  1.715457220001 = 2*cos(11*pi/64) = sqrt(2+sqrt(2-sqrt(2-sqrt(2-sqrt(2)))))
  1.727945712243 = 2*cos(43*pi/256) = sqrt(2+sqrt(2-sqrt(2-sqrt(2-sqrt(2-sqrt(2-sqrt(2)))))))
  1.732050807569 = 2*cos(pi/6) = sqrt(3)
  1.740173982217 = 2*cos(21*pi/128) = sqrt(2+sqrt(2-sqrt(2-sqrt(2-sqrt(2-sqrt(2))))))
  1.744992014146 = 2*cos(13*pi/80) = sqrt(4+sqrt(8-sqrt(2)+sqrt(10)-2*sqrt(5+sqrt(5))))/sqrt(2)
  1.747244781293 = 2*cos(11*pi/68) = sqrt(8+sqrt(2)*sqrt(17+sqrt(17)-sqrt(34+2*sqrt(17))-2*sqrt(17-3*sqrt(17)-sqrt(34+2*sqrt(17))+2*sqrt(34-2*sqrt(17)))))/2
  1.748180683253 = 2*cos(31*pi/192) = sqrt(2+sqrt(2-sqrt(2-sqrt(2-sqrt(2-sqrt(3))))))
  1.752140188391 = 2*cos(41*pi/256) = sqrt(2+sqrt(2-sqrt(2-sqrt(2-sqrt(2-sqrt(2+sqrt(2)))))))
  1.762024388572 = 2*cos(8*pi/51) = (1+sqrt(17)+sqrt(34+2*sqrt(17))+2*sqrt(17-3*sqrt(17)+sqrt(34+2*sqrt(17))-2*sqrt(34-2*sqrt(17))))/16 + sqrt(6)*sqrt(17+sqrt(17)-sqrt(34+2*sqrt(17))-2*sqrt(17-3*sqrt(17)-sqrt(34+2*sqrt(17))+2*sqrt(34-2*sqrt(17))))/8
  1.763842528697 = 2*cos(5*pi/32) = sqrt(2+sqrt(2-sqrt(2-sqrt(2))))
  1.775279240806 = 2*cos(39*pi/256) = sqrt(2+sqrt(2-sqrt(2-sqrt(2+sqrt(2-sqrt(2+sqrt(2)))))))
  1.779032150844 = 2*cos(29*pi/192) = sqrt(2+sqrt(2-sqrt(2-sqrt(2+sqrt(2-sqrt(3))))))
  1.782013048377 = 2*cos(3*pi/20) = (sqrt(10)-sqrt(2)+2*sqrt(5+sqrt(5)))/4
  1.786448602391 = 2*cos(19*pi/128) = sqrt(2+sqrt(2-sqrt(2-sqrt(2+sqrt(2-sqrt(2))))))
  1.790326582710 = 2*cos(5*pi/34) = sqrt(2)*sqrt(17+sqrt(17)+sqrt(34+2*sqrt(17))-2*sqrt(17-3*sqrt(17)+sqrt(34+2*sqrt(17))-2*sqrt(34-2*sqrt(17))))/4
  1.793745483065 = 2*cos(7*pi/48) = sqrt(2+sqrt(2-sqrt(2-sqrt(3))))
  1.797348931388 = 2*cos(37*pi/256) = sqrt(2+sqrt(2-sqrt(2-sqrt(2+sqrt(2-sqrt(2-sqrt(2)))))))
  1.807978586247 = 2*cos(9*pi/64) = sqrt(2+sqrt(2-sqrt(2-sqrt(2+sqrt(2)))))
  1.816286347650 = 2*cos(11*pi/80) = sqrt(4+sqrt(8-sqrt(2)-sqrt(10)+2*sqrt(5-sqrt(5))))/sqrt(2)
  1.816930543639 = 2*cos(7*pi/51) = (-1+sqrt(17)+sqrt(34-2*sqrt(17))-2*sqrt(17+3*sqrt(17)-sqrt(34-2*sqrt(17))-2*sqrt(34+2*sqrt(17))))/16 + sqrt(6)*sqrt(17-sqrt(17)+sqrt(34-2*sqrt(17))+2*sqrt(17+3*sqrt(17)+sqrt(34-2*sqrt(17))+2*sqrt(34+2*sqrt(17))))/8
  1.818335966181 = 2*cos(35*pi/256) = sqrt(2+sqrt(2-sqrt(2-sqrt(2+sqrt(2+sqrt(2-sqrt(2)))))))
  1.821727649842 = 2*cos(13*pi/96) = sqrt(2+sqrt(2-sqrt(2-sqrt(2+sqrt(3)))))
  1.827090915285 = 2*cos(2*pi/15) = (1+sqrt(5)+sqrt(30-6*sqrt(5)))/4
  1.828419511407 = 2*cos(17*pi/128) = sqrt(2+sqrt(2-sqrt(2-sqrt(2+sqrt(2+sqrt(2))))))
  1.829587736976 = 2*cos(9*pi/68) = sqrt(8+sqrt(2)*sqrt(17-sqrt(17)-sqrt(34-2*sqrt(17))+2*sqrt(17+3*sqrt(17)-sqrt(34-2*sqrt(17))-2*sqrt(34+2*sqrt(17)))))/2
  1.834988992895 = 2*cos(25*pi/192) = sqrt(2+sqrt(2-sqrt(2-sqrt(2+sqrt(2+sqrt(3))))))
  1.838227703380 = 2*cos(33*pi/256) = sqrt(2+sqrt(2-sqrt(2-sqrt(2+sqrt(2+sqrt(2+sqrt(2)))))))
  1.847759065023 = 2*cos(pi/8) = sqrt(2+sqrt(2))
  1.857012160946 = 2*cos(31*pi/256) = sqrt(2+sqrt(2+sqrt(2-sqrt(2+sqrt(2+sqrt(2+sqrt(2)))))))
  1.860034447368 = 2*cos(23*pi/192) = sqrt(2+sqrt(2+sqrt(2-sqrt(2+sqrt(2+sqrt(3))))))
  1.864944458809 = 2*cos(2*pi/17) = (-1+sqrt(17)+sqrt(34-2*sqrt(17))+2*sqrt(17+3*sqrt(17)-sqrt(34-2*sqrt(17))-2*sqrt(34+2*sqrt(17))))/8
  1.865985597669 = 2*cos(15*pi/128) = sqrt(2+sqrt(2+sqrt(2-sqrt(2+sqrt(2+sqrt(2))))))
  1.867160852994 = 2*cos(7*pi/60) = (sqrt(2)+sqrt(6)+sqrt(10)+sqrt(30)-2*(1-sqrt(3))*(sqrt(5-sqrt(5))))/8
  1.871811853515 = 2*cos(11*pi/96) = sqrt(2+sqrt(2+sqrt(2-sqrt(2+sqrt(3)))))
  1.874678023825 = 2*cos(29*pi/256) = sqrt(2+sqrt(2+sqrt(2-sqrt(2+sqrt(2+sqrt(2-sqrt(2)))))))
  1.876382671845 = 2*cos(9*pi/80) = sqrt(4+sqrt(8+sqrt(2)+sqrt(10)-2*sqrt(5-sqrt(5))))/sqrt(2)
  1.883088130366 = 2*cos(7*pi/64) = sqrt(2+sqrt(2+sqrt(2-sqrt(2+sqrt(2)))))
  1.891214650761 = 2*cos(27*pi/256) = sqrt(2+sqrt(2+sqrt(2-sqrt(2+sqrt(2-sqrt(2-sqrt(2)))))))
  1.893860258990 = 2*cos(5*pi/48) = sqrt(2+sqrt(2+sqrt(2-sqrt(3))))
  1.896321295182 = 2*cos(7*pi/68) = sqrt(8+sqrt(2)*sqrt(17+sqrt(17)-sqrt(34+2*sqrt(17))+2*sqrt(17-3*sqrt(17)-sqrt(34+2*sqrt(17))+2*sqrt(34-2*sqrt(17)))))/2
  1.899056361186 = 2*cos(13*pi/128) = sqrt(2+sqrt(2+sqrt(2-sqrt(2+sqrt(2-sqrt(2))))))
  1.902113032590 = 2*cos(pi/10) = (sqrt(10+2*sqrt(5)))/2
  1.904125355428 = 2*cos(19*pi/192) = sqrt(2+sqrt(2+sqrt(2-sqrt(2+sqrt(2-sqrt(3))))))
  1.905884000854 = 2*cos(5*pi/51) = (-1+sqrt(17)-sqrt(34-2*sqrt(17))+2*sqrt(17+3*sqrt(17)+sqrt(34-2*sqrt(17))+2*sqrt(34+2*sqrt(17))))/16 + sqrt(6)*sqrt(17-sqrt(17)-sqrt(34-2*sqrt(17))+2*sqrt(17+3*sqrt(17)-sqrt(34-2*sqrt(17))-2*sqrt(34+2*sqrt(17))))/8
  1.906612080708 = 2*cos(25*pi/256) = sqrt(2+sqrt(2+sqrt(2-sqrt(2+sqrt(2-sqrt(2+sqrt(2)))))))
  1.913880671464 = 2*cos(3*pi/32) = sqrt(2+sqrt(2+sqrt(2-sqrt(2))))
  1.920861038831 = 2*cos(23*pi/256) = sqrt(2+sqrt(2+sqrt(2-sqrt(2-sqrt(2-sqrt(2+sqrt(2)))))))
  1.923123595366 = 2*cos(17*pi/192) = sqrt(2+sqrt(2+sqrt(2-sqrt(2-sqrt(2-sqrt(3))))))
  1.923651286346 = 2*cos(3*pi/34) = sqrt(2)*sqrt(17+sqrt(17)+sqrt(34+2*sqrt(17))+2*sqrt(17-3*sqrt(17)+sqrt(34+2*sqrt(17))-2*sqrt(34-2*sqrt(17))))/4
  1.924910472907 = 2*cos(7*pi/80) = sqrt(4+sqrt(8+sqrt(2)-sqrt(10)+2*sqrt(5+sqrt(5))))/sqrt(2)
  1.927552131591 = 2*cos(11*pi/128) = sqrt(2+sqrt(2+sqrt(2-sqrt(2-sqrt(2-sqrt(2))))))
  1.931851652578 = 2*cos(pi/12) = sqrt(2+sqrt(3))
  1.933952942090 = 2*cos(21*pi/256) = sqrt(2+sqrt(2+sqrt(2-sqrt(2-sqrt(2-sqrt(2-sqrt(2)))))))
  1.939593872070 = 2*cos(4*pi/51) = (1+sqrt(17)-sqrt(34+2*sqrt(17))+2*sqrt(17-3*sqrt(17)-sqrt(34+2*sqrt(17))+2*sqrt(34-2*sqrt(17))))/16 + sqrt(6)*sqrt(17+sqrt(17)+sqrt(34+2*sqrt(17))+2*sqrt(17-3*sqrt(17)+sqrt(34+2*sqrt(17))-2*sqrt(34-2*sqrt(17))))/8
  1.940062506389 = 2*cos(5*pi/64) = sqrt(2+sqrt(2+sqrt(2-sqrt(2-sqrt(2)))))
  1.944739840795 = 2*cos(3*pi/40) = sqrt(8-sqrt(2)+sqrt(10)+2*sqrt(5+sqrt(5)))/2
  1.945879904411 = 2*cos(19*pi/256) = sqrt(2+sqrt(2+sqrt(2-sqrt(2-sqrt(2+sqrt(2-sqrt(2)))))))
  1.946876108721 = 2*cos(5*pi/68) = sqrt(8+sqrt(2)*sqrt(17+sqrt(17)+sqrt(34+2*sqrt(17))-2*sqrt(17-3*sqrt(17)+sqrt(34+2*sqrt(17))-2*sqrt(34-2*sqrt(17)))))/2
  1.947753958555 = 2*cos(7*pi/96) = sqrt(2+sqrt(2+sqrt(2-sqrt(2-sqrt(3)))))
  1.951404260077 = 2*cos(9*pi/128) = sqrt(2+sqrt(2+sqrt(2-sqrt(2-sqrt(2+sqrt(2))))))
  1.954923949887 = 2*cos(13*pi/192) = sqrt(2+sqrt(2+sqrt(2-sqrt(2-sqrt(2+sqrt(3))))))
  1.956295201468 = 2*cos(pi/15) = (sqrt(5)-1+sqrt(30+6*sqrt(5)))/4
  1.956634741439 = 2*cos(17*pi/256) = sqrt(2+sqrt(2+sqrt(2-sqrt(2-sqrt(2+sqrt(2+sqrt(2)))))))
  1.961570560806 = 2*cos(pi/16) = sqrt(2+sqrt(2+sqrt(2)))
  1.965946199368 = 2*cos(pi/17) = (1-sqrt(17)+sqrt(34-2*sqrt(17))+2*sqrt(17+3*sqrt(17)+sqrt(34-2*sqrt(17))+2*sqrt(34+2*sqrt(17))))/8
  1.966210974862 = 2*cos(15*pi/256) = sqrt(2+sqrt(2+sqrt(2+sqrt(2-sqrt(2+sqrt(2+sqrt(2)))))))
  1.967692011854 = 2*cos(11*pi/192) = sqrt(2+sqrt(2+sqrt(2+sqrt(2-sqrt(2+sqrt(3))))))
  1.970555284778 = 2*cos(7*pi/128) = sqrt(2+sqrt(2+sqrt(2+sqrt(2-sqrt(2+sqrt(2))))))
  1.973286664170 = 2*cos(5*pi/96) = sqrt(2+sqrt(2+sqrt(2+sqrt(2-sqrt(3)))))
  1.974602836316 = 2*cos(13*pi/256) = sqrt(2+sqrt(2+sqrt(2+sqrt(2-sqrt(2+sqrt(2-sqrt(2)))))))
  1.975376681190 = 2*cos(pi/20) = (sqrt(2)+sqrt(10)+2*(sqrt(5-sqrt(5))))/4
  1.978353019930 = 2*cos(3*pi/64) = sqrt(2+sqrt(2+sqrt(2+sqrt(2-sqrt(2)))))
  1.980820861750 = 2*cos(3*pi/68) = sqrt(8+sqrt(2)*sqrt(17+sqrt(17)+sqrt(34+2*sqrt(17))+2*sqrt(17-3*sqrt(17)+sqrt(34+2*sqrt(17))-2*sqrt(34-2*sqrt(17)))))/2
  1.981805270856 = 2*cos(11*pi/256) = sqrt(2+sqrt(2+sqrt(2+sqrt(2-sqrt(2-sqrt(2-sqrt(2)))))))
  1.982889722748 = 2*cos(pi/24) = sqrt(2+sqrt(2+sqrt(3)))
  1.984841019344 = 2*cos(2*pi/51) = (1+sqrt(17)+sqrt(34+2*sqrt(17))-2*sqrt(17-3*sqrt(17)+sqrt(34+2*sqrt(17))-2*sqrt(34-2*sqrt(17))))/16 + sqrt(6)*sqrt(17+sqrt(17)-sqrt(34+2*sqrt(17))+2*sqrt(17-3*sqrt(17)-sqrt(34+2*sqrt(17))+2*sqrt(34-2*sqrt(17))))/8
  1.984959069197 = 2*cos(5*pi/128) = sqrt(2+sqrt(2+sqrt(2+sqrt(2-sqrt(2-sqrt(2))))))
  1.986136913910 = 2*cos(3*pi/80) = sqrt(4+sqrt(8-sqrt(2)+sqrt(10)+2*sqrt(5+sqrt(5))))/sqrt(2)
  1.986895558039 = 2*cos(7*pi/192) = sqrt(2+sqrt(2+sqrt(2+sqrt(2-sqrt(2-sqrt(3))))))
  1.987813940005 = 2*cos(9*pi/256) = sqrt(2+sqrt(2+sqrt(2+sqrt(2-sqrt(2-sqrt(2+sqrt(2)))))))
  1.989043790737 = 2*cos(pi/30) = (sqrt(15)+sqrt(3)+sqrt(10-2*sqrt(5)))/4
  1.990369453344 = 2*cos(pi/32) = sqrt(2+sqrt(2+sqrt(2+sqrt(2))))
  1.991468352590 = 2*cos(pi/34) = sqrt(2)*sqrt(17-sqrt(17)+sqrt(34-2*sqrt(17))+2*sqrt(17+3*sqrt(17)+sqrt(34-2*sqrt(17))+2*sqrt(34+2*sqrt(17))))/4
  1.992625224366 = 2*cos(7*pi/256) = sqrt(2+sqrt(2+sqrt(2+sqrt(2+sqrt(2-sqrt(2+sqrt(2)))))))
  1.993310478618 = 2*cos(5*pi/192) = sqrt(2+sqrt(2+sqrt(2+sqrt(2+sqrt(2-sqrt(3))))))
  1.993834667466 = 2*cos(pi/40) = sqrt(8+sqrt(2)+sqrt(10)+2*sqrt(5-sqrt(5)))/2
  1.994580913357 = 2*cos(3*pi/128) = sqrt(2+sqrt(2+sqrt(2+sqrt(2+sqrt(2-sqrt(2))))))
  1.995717846477 = 2*cos(pi/48) = sqrt(2+sqrt(2+sqrt(2+sqrt(3))))
  1.996206657474 = 2*cos(pi/51) = (-1-sqrt(17)+sqrt(34+2*sqrt(17))+2*sqrt(17-3*sqrt(17)-sqrt(34+2*sqrt(17))+2*sqrt(34-2*sqrt(17))))/16 + sqrt(6)*sqrt(17+sqrt(17)+sqrt(34+2*sqrt(17))-2*sqrt(17-3*sqrt(17)+sqrt(34+2*sqrt(17))-2*sqrt(34-2*sqrt(17))))/8
  1.996236225800 = 2*cos(5*pi/256) = sqrt(2+sqrt(2+sqrt(2+sqrt(2+sqrt(2-sqrt(2-sqrt(2)))))))
  1.997259069509 = 2*cos(pi/60) = (sqrt(2)-sqrt(6)-sqrt(10)+sqrt(30)+2*(1+sqrt(3))*(sqrt(5+sqrt(5))))/8
  1.997590912410 = 2*cos(pi/64) = sqrt(2+sqrt(2+sqrt(2+sqrt(2+sqrt(2)))))
  1.997865949605 = 2*cos(pi/68) = sqrt(8+sqrt(2)*sqrt(17-sqrt(17)+sqrt(34-2*sqrt(17))+2*sqrt(17+3*sqrt(17)+sqrt(34-2*sqrt(17))+2*sqrt(34+2*sqrt(17)))))/2
  1.998458072481 = 2*cos(pi/80) = sqrt(4+sqrt(8+sqrt(2)+sqrt(10)+2*sqrt(5-sqrt(5))))/sqrt(2)
  1.998644769177 = 2*cos(3*pi/256) = sqrt(2+sqrt(2+sqrt(2+sqrt(2+sqrt(2+sqrt(2-sqrt(2)))))))
  1.998929174953 = 2*cos(pi/96) = sqrt(2+sqrt(2+sqrt(2+sqrt(2+sqrt(3)))))
  1.999397637392 = 2*cos(pi/128) = sqrt(2+sqrt(2+sqrt(2+sqrt(2+sqrt(2+sqrt(2))))))
  1.999732275819 = 2*cos(pi/192) = sqrt(2+sqrt(2+sqrt(2+sqrt(2+sqrt(2+sqrt(3))))))
  1.999849403678 = 2*cos(pi/256) = sqrt(2+sqrt(2+sqrt(2+sqrt(2+sqrt(2+sqrt(2+sqrt(2)))))))

In deriving that list, the famous identity cos(2φ)=2 cos²(φ)–1 is most helpful for cases with even denominators. For prime denominators, de Moivre’s formula (cos(2π/n) + i*sin(2π/n))ⁿ = 1 can be used (read: The n-th power of an n-th root of unity is one). Other cases are accessible by using the standard addition theorem cos(2π(n+m)/(n*m)) = cos(2π/n)*cos(2π/m) – sin(2π/n)*sin(2π/m). As usual, the worst part is the simplification of the resulting radical expression. Note that cosine values for angles with a given denominator differ only in the signs of the various terms and need not to be derived afresh.

This page was written by Gernot Katzer

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