Max Brückner photographed 146 paper polyhedra he had built himself and printed
them as collotype plates — the first major work to use photography to show what
mathematical objects look like. This page puts three things side by side: the
plates as they were printed, the pages on which he describes each figure, and the
same solid rebuilt from scratch by the stellation engine. Where he derives a solid
from what he calls the complete figure of the planes, that figure is drawn
live, and it can be clicked.
Scan: the University of Toronto copy at the Internet Archive. Text and plate
references were read out of the scan itself, not taken from secondary sources.
The booknot open
Every read the page and see the plate button below opens the scan
here. The printed text runs to p. 227; the plates follow it.
1 · What the book is
Vielecke und Vielflache. Theorie und Geschichte — polygons and polyhedra,
theory and history — was published by B. G. Teubner of Leipzig in 1900. The title
page gives the author's job: Dr. Max Brückner, Oberlehrer am Gymnasium zu
Bautzen. He was a schoolmaster, and remained one; the book was written in the
evenings.
Its own subtitle describes the illustrations exactly: Mit 7 lithographierten
und 5 Lichtdruck-Doppeltafeln sowie vielen Figuren im Text — with seven
lithographed and five collotype double-plates, and many figures in the text. The
lithographs, Plates I–VII, carry the construction figures, the nets and the
face diagrams. The collotypes, Plates VIII–XII, carry the photographs. Because
each is a double plate, they run across ten sheets and show 146 models.
Collotype — Lichtdruck — is a continuous-tone photographic process printed
from a gelatine plate, with no halftone screen. That is why the plates have their
soft, almost sculptural grey, and why they read as photographs rather than as
illustrations. The printer's credit under each is Lichtdruck von Römmler
& Jonas, Dresden.
The text runs to 227 printed pages in six parts, A to F. Part F, Die
besonderen Vielflache höherer Art — the special polyhedra of higher kind — is
where the star polyhedra, the compounds and the stellations are, and it is the
part this page is concerned with.
2 · The twelve plates
Any plate can be opened in the reader above. The models on the collotype plates
are arranged by size rather than by subject — apart from a rough split
that keeps the isohedral models off the isogonal ones' plates. Brückner explains in
a footnote that distributing them by whatever size each happened to come out at was
necessary to avoid wasting plate space, and apologises for the flipping back and
forth it forces on the reader. In the preface he separately thanks the printers and
the publisher for absorbing the not inconsiderable cost of producing them.
There is also a clickable edition of the five
photographic plates: every one of the 146 models has been identified where
the book's text or the modern literature allows it, and the ones the engine can
rebuild are live — click a model on the plate and it is rebuilt beneath, with
its other names and its history.
3 · §153 — the complete figure of the planes of the icosahedron
This is the section that connects the book to the program. Brückner's own
description of the construction, on p. 206:
Bringt man die Ebene einer Fläche des Ikosaeders mit den Ebenen aller übrigen
zum Schnitt, so entsteht die schon früher betrachtete Fig. 17 Taf. II. … Die
Ebene 20 schneidet die Zeichenebene in der unendlichweiten Geraden. Die auf
demselben konzentrischen Kreise um den Mittelpunkt der Fläche 1 liegenden Punkte
tragen dieselbe Ziffer als Bezeichnung.
Bring the plane of one face of the icosahedron into intersection with the planes
of all the others and Fig. 17 of Plate II results. … Plane 20 meets the drawing
plane in the line at infinity. Points lying on the same concentric circle about
the centre of face 1 carry the same digit as their label.
— Vielecke und Vielflache, §153, p. 206.
Three things in that passage are exactly what the program computes. Plane 20 is
the face opposite face 1: it is parallel, so it never meets the drawing plane,
which is why twenty planes produce only eighteen lines. The concentric rings of
intersection points are the layers — Brückner's digits 1, 2, 3 … are the numbers
down the left of the Cells table. And the whole figure is the panel the program
calls the stellation diagram.
Plate II, lithographed. Fig. 17 is the large star at the lower right —
the complete figure of the icosahedron's planes, and the drawing §153 refers to.
Fig. 16, which §153 names as the dodecahedron's, is on the facing sheet.
Click to enlarge.
The same arrangement, computed. In the first setting the program draws only the
plane traces, at full width, with the original face tinted at the centre — the
engraved look of the plate. In the second it fills each bounded region and tints
it by which shell the cell behind it belongs to, which is the working view. The
traces are the same lines either way, and there are eighteen of them, for
the reason Brückner gives above.
Drag to pan, scroll to zoom, double-click to reset. shift-click
a region to add the cell behind it, ctrl-click to
remove it.
Brückner reads solids straight off this figure, and §§153–155 do it three times
over: the points labelled 6 and 7 give the boundary faces of two solids on
Plate VIII and Plate IX, and the nine points labelled 8 give the face of the solid
on Plate XI. Those derivations are the same operation as choosing cells here.
4 · The compounds, plate by plate
Plate IX is the compound plate, and it is the one that travelled furthest: Escher
cited it twice by number in his own handwriting. Each entry below gives Brückner's
plate and figure number, what he says about it and where, and the same solid built
from the icosahedron's cells.
… die Grenzfläche eines zugleich gleicheckigen und gleichflächigen, aber
diskontinuierlichen Polyeders, nämlich eines Systems von fünf konzentrischen
Oktaedern, Fig. 6 Taf. IX.
… the boundary face of a polyhedron at once isogonal and isohedral, but
discontinuous — namely a system of five concentric octahedra. §153, p. 206.
"Discontinuous" is Brückner's word for what would now be called a compound:
a figure that falls apart into pieces. He obtains it from the complete
figure, by joining the points labelled 3.
In cells it is everything out to the third shell, {0,1,2}. The
trap is that the shell underneath is invisible from outside: taking only
{0,2} gives the same silhouette and a hollow interior.
… eines Systems von fünf Tetraedern. Die äussere Hülle ist ein Dodekaeder.
Diese beiden Systeme, von denen das eine in Fig. 11 Taf. IX dargestellt ist,
die sich zu einander …
… a system of five tetrahedra. The outer hull is a dodecahedron. These
two systems, of which one is shown in Fig. 11 of Plate IX, which stand to one
another [as mirror images]. §153, p. 206.
Brückner states the two facts that matter. The outer hull is a dodecahedron:
in the model below, the twenty outermost vertices are the vertices of one.
And the figure comes in two systems, mirror images that no rotation
interchanges — which is why it is the standard example of chirality in this
subject.
In cells, everything out to shell e plus one half of the single
chiral bundle in shell f. Taking both halves gives the compound of ten
tetrahedra, which Brückner photographs as Fig. 3 on the same plate.
Die fünf Tetraeder und die zehn Tetraeder ohne das Dodekaeder zeigen
Fig. 11 und Fig. 3 Taf. IX.
The five tetrahedra and the ten tetrahedra, without the dodecahedron,
are shown by Fig. 11 and Fig. 3 of Plate IX. Footnote, p. 131.
The qualification "without the dodecahedron" is a model-maker's note: the
same compounds are often shown inscribed in a dodecahedron, and his models
omit it so that the tetrahedra can be seen. The footnote on the same page
sends the reader to Fig. 6 for the octahedra.
The five-tetrahedron and ten-tetrahedron compounds differ in exactly one
place in the cell notation, which is the point of the third button on the
previous figure: both take every cell out to shell e, and then one takes
half of the chiral bundle in shell f while the other takes all of it.
Taf. IX, 23
Three concentric cubes — the one Escher wrote down
… so entsteht ein System von drei konzentrischen Hexaedern, Fig. 23 Taf.
IX, dessen äussere Hülle eine bestimmte Varietät …
… there arises a system of three concentric hexahedra, Fig. 23 of Plate
IX, whose outer hull is a certain variety … p. 188.
Escher made a coloured-pencil drawing of this compound and annotated it, in
his own hand, Brückner, Table IX, no. 23. He did the same for the
compound of five tetrahedra, citing Table IX, no. 11. Those two
annotations are why the dependence of Escher's polyhedra on this book is
documented rather than inferred. The compound of three cubes went on to
become the left-hand tower of Waterfall (1961).
It is not a stellation and the program cannot produce it. A stellation must
keep its faces in the planes of the solid it started from, and the cube's
six planes bound nothing but the cube — the cube is the one Platonic solid
with no stellation at all. Three cubes at angles need twenty-four planes that
belong to no single starting solid.
Es lässt sich aber auch ein bisher noch nicht beschriebenes gleichflächiges
Polyeder höherer Art aus dieser vollständigen Ikosaederfigur ableiten.
But an isohedral polyhedron of higher kind not hitherto described can
also be derived from this complete icosahedron figure. §153, p. 207.
He then gives the recipe: join each of the three points 7 to the two points 6
on the opposite edge of the triangle 7,7,7, and add every segment between
points 6 parallel to that triangle's edges. The result is a nine-sided figure
of the fourth kind, and twenty of them bound the solid photographed as
Fig. 20 — the tall many-spiked model at the centre of the second sheet of
Plate IX. Its twelve five-edged vertices are an icosahedron's; its twenty
six-edged vertices are the points 6.
This is the clearest demonstration in the book of what the complete figure is
for. The construction is carried out on the very diagram drawn in
§3 above; readers who wish to follow it can zoom into the live figure and
count rings outward. The corresponding cell selection is not identified here,
because Brückner names the boundary polygon rather than the solid, and a
confident identification needs more than a plausible match.
5 · The stellations of the icosahedron
The book shows ten stellations of the icosahedron, several of which Brückner was
the first to build. Two of them are on Plate XI and are worth taking one at a
time, because they sit at opposite ends of the same table: the outermost cell of
all, and the last of the four regular star polyhedra.
Top centre of the plate, and unmistakable: a body of long spikes in every
direction. It is the stellation that takes every cell — the point at
which the twenty planes have nothing further to give. Brückner derives its
boundary face in §155 from the nine points labelled 8 in the complete figure,
counting twenty-seven double points on it.
Rebuilt below it comes out at 92 vertices and 180 triangles, which are the
figures the modern literature gives. The name "echidnahedron", after the
spiny anteater, is a twentieth-century coinage — Andrew Hume, 1995 — and no
part of Brückner's vocabulary.
Dieses komplizierteste der vier regulären Polyeder höherer Art zeigt
Fig. 24 Taf. VII … und Fig. 24 Taf. XI.
This most complicated of the four regular polyhedra of higher kind is
shown by Fig. 24 of Plate VII … and Fig. 24 of Plate XI. p. 169.
Poinsot's solid of 1809, the only one of the four Kepler–Poinsot star
polyhedra that is a stellation of the icosahedron rather than the
dodecahedron. Brückner gives it twice: a lithographed perspective drawing on
Plate VII, with the boundary face lettered abc on the solid itself, and
the photographed model on the collotype Plate XI — the two halves of his
method, the drawing and the object.
It takes seven of the icosahedron's eight shells. Its twelve outermost
vertices sit at φ³ ≈ 4.236 times the core's circumradius, which the readout
below can be used to check.
Fourth row of the plate, first model. Brückner discusses it on p. 191 as the
polar solid of a figure bounded by hexagons of the second kind lying in the
planes of an octahedron and by quadrilaterals in the planes of a rhombic
dodecahedron. The printed text says vierzehn, fourteen, but the face
symbol in the same sentence gives twelve, and a rhombic dodecahedron has
twelve planes; the book has a misprint here.
It is now usually called Escher's solid, from the right-hand tower of
Waterfall. It is also a compound of three flattened octahedra — and
that near-coincidence produced a real artistic decision: a surviving
preliminary study for Stars (1948) uses this solid, where the finished
print uses three regular octahedra, Brückner's Plate VIII no. 12.
The two look almost identical. One is a stellation and the other is not, and
Escher tried both before choosing.
As cells: the rhombic dodecahedron plus its first shell. The result has 26
vertices, 48 triangles, and exactly twice the volume of the core.
§156
The complete figure of the rhombic triacontahedron
buildable
Having done the icosahedron in §153, Brückner does the same for the rhombic
triacontahedron in §156, and derives from it the solids of §§157–158. He
states his reason plainly at the head of the run: of all the isohedral solids
of the first kind, only two need the treatment — das Ikosaeder und das
Rhombentriakontaeder — because everything else either reduces to them or
has been dealt with already.
It is a much larger arrangement. Thirty planes rather than twenty give
thirteen shells and several thousand cells, which is why the figure below
takes a moment to appear and why the program caps the depth by default.
7 · How the plates were made
The single most useful footnote in the book is note 2 on p. 183, and it explains
why the plates look the way they do.
On the isogonal models, faces with the same number of edges were painted
the same oil colour, so that the configuration could be taken in at a glance — a
convention Brückner says he adopted from models Hess exhibited at the mathematical
exhibition in Munich in 1893. Coloured models, however, photographed badly for
collotype. So every one of those had to be painted over white, and to stop the
white from shining it was applied as a coat of Schlemmkreide, levigated
chalk. Once chalked, the surface would no longer take the drawn edge lines.
The uniform ghostly matte white of the famous plates is therefore not an
aesthetic. It is what a 1900 camera required. The models standing in Brückner's
room were colour-coded; the models in the book were whitewashed for the
photographer.
The same footnote records that some of the polyhedra described in the text are not
pictured at all — partly because they were too complicated, and partly because, in
his own words, the author had finally run out of the thing he most needed:
patience. He adds that he hopes to supply the missing ones later.
Plate VIII, first sheet. Twenty whitewashed models on five shelves, each
casting its own shadow. No. 3, top row, is the compound of two cubes on a
common 3-fold axis; both it and no. 12 appear in Escher's Stars.
Plate IX, second sheet. No. 20 is the solid he says had not been described
before; no. 23 is the compound of three cubes that Escher copied out by hand.
Brückner gave over two hundred of these models to the Mathematical Institute at
Heidelberg in 1930–31, and the university awarded him an honorary doctorate.
There is no record of what became of them. They survived the First World War;
German scholarship on model collections reports the Heidelberg holdings as lost,
and no Brückner model is known to survive anywhere. The photographs are what is
left, which is why a page like this one is possible at all.
Sources and caveats
Every plate and figure reference on this page was read out of the scan itself, and
every German passage quoted was transcribed from the same source and translated
here; where the optical character recognition was doubtful the wording was checked
against the page image. The page numbers are Brückner's printed ones. The
correspondence between printed page and scan image was taken from the Archive's
own pagination data, and the plate sheets were identified by eye from their
captions.
One identification on this page rests on secondary literature rather than on the
book's own words: that Escher's two annotated drawings cite Plate IX nos. 11 and 23,
which is from George Hart's study of the book. Plate VIII no. 12 needs no secondary
source — on p. 188, in the sentence following the three-cubes passage quoted above,
Brückner writes Das System der polar zugeordneten drei Oktaeder zeigt Fig. 12
Taf. VIII. The reading of Plate XI no. 14 as the final stellation of the
icosahedron is Hart's, and is consistent with what Brückner derives in §155 and with
the model as photographed.
Sources
Max Brückner, Vielecke und Vielflache: Theorie und Geschichte,
B. G. Teubner, Leipzig, 1900. Scan of the University of Toronto copy at the
Internet Archive;
also at the
University of Michigan.
Not in copyright.
George W. Hart, "Max Brückner's Wunderkammer of Paper Polyhedra",
Bridges 2019 — the modern account of the book, its reception, and its
route into Escher's work.
Frank Etwein, "Ein Leben für die Polyeder — der Oberlehrer Max Brückner und
seine Modelle", Mathematische Semesterberichte 66 (2019), 15–30.
Edmund Hess, Über die zugleich gleicheckigen und gleichflächigen
Polyeder, Kassel, 1876 — Brückner's "Hess III", the source he credits for
the complete-figure method.