U.S. patent application number 10/275159 was filed with the patent office on 2003-07-24 for structural elements and tile sets.
Invention is credited to Geissler, Bernhard.
Application Number | 20030136069 10/275159 |
Document ID | / |
Family ID | 7640690 |
Filed Date | 2003-07-24 |
United States Patent
Application |
20030136069 |
Kind Code |
A1 |
Geissler, Bernhard |
July 24, 2003 |
Structural elements and tile sets
Abstract
The invention relates to sets of structural elements
representing equilateral irregular pentagonal structures with
angles of 36.degree., 108.degree., 108.degree., 36.degree. and
252.degree., which enable a plurality of patterns that can cover a
plane periodically or non-periodically. Modified structural
elements in which each side of the equilateral, irregular pentagon
is replaced by point-symmetrical lines also cover the plane. Sets
of said structural elements can be used for tile applications, tile
games such as puzzles, intarsia products, jewelry and the
production of fabrics and wallpaper.
Inventors: |
Geissler, Bernhard; (Munich,
DE) |
Correspondence
Address: |
FOLEY AND LARDNER
SUITE 500
3000 K STREET NW
WASHINGTON
DC
20007
US
|
Family ID: |
7640690 |
Appl. No.: |
10/275159 |
Filed: |
November 4, 2002 |
PCT Filed: |
May 4, 2001 |
PCT NO: |
PCT/EP01/05058 |
Current U.S.
Class: |
52/311.2 |
Current CPC
Class: |
B44F 3/00 20130101; A63F
2009/0697 20130101; B44F 11/04 20130101; B44C 3/123 20130101; A63F
2009/0681 20130101; A63F 9/06 20130101; E04F 15/02 20130101; A63F
2009/0695 20130101 |
Class at
Publication: |
52/311.2 |
International
Class: |
E04F 013/00; E04F
015/00; E04F 019/00 |
Claims
1. Set of structural elements which comprises at least one subset
UA of congruent structural elements SA with a circumference derived
from an equilateral polygon, characterized in that the
circumference of all structural elements SA consists of five lines
LF, which connect in a closed way the comers of an equilateral,
irregular pentagon (GUF), wherein the angles of the equilateral
irregular pentagon are 36.degree., 108.degree., 108.degree.,
36.degree. and 252.degree..
2. Set of structural elements in accordance with claim 1,
characterized in that the lines LF are congruent and selected from
straight lines GL centrally symmetrical non-straight lines ZL
mirror symmetrical non-straight lines SL, and in that the
connecting lines ZL and SL do not intersect, preferably also do not
touch except in the corners of the GUF.
3. Set of structural elements in accordance with claim 2,
characterized in that these comprise at least one further subset UB
of second structural elements SB differing from the structural
elements SA but being complimentary to the subgroup UA in such a
way that the subgroups UA and UB can be juxtaposed so that they
cover a portion of the plane without gaps.
4. Set of structural elements in accordance with claim 3,
characterized in that the structural elements SA have a
circumference which results from lines LF which consist of two
sections LC of equal length forming an angle of 144.degree. between
them and forming an irregular equilateral decagon (GUZ) and in that
the structural elements SB have the circumference of an equilateral
irregular hexagon (GUS) with six line elements (LS) of the same
length as LZ and the internal angles of 72.degree., 216.degree.,
72.degree., 72.degree., 216.degree., 72.degree..
5. Set of structural elements in accordance with one of the
preceding claims characterized in that the circumference of the
structural elements SA respectively SB are generated by connecting
the corners of the GUF or respectively the GUZ or respectively the
GUS by congruent, centrally symmetrically, non-straight lines, or
by congruent mirror symmetrical non-straight lines which do not
intersect
6. Structural element, in particular placement game element, tile
or inlay part, characterized in that the structural element has a
circumference which connects the cornets of an equilateral,
irregular pentagon (FIG. 1) with the angles 36.degree.,
108.degree., 108.degree., 36.degree., 252.degree. by the lines (b)
or (c), or the corners of an equilateral decagon (FIG. 11b) with
the angles 288.degree., 144.degree., 72.degree., 144.degree.,
144.degree., 144.degree., 144.degree., 144.degree. by the lines
(a), (b) or (c), or the corners of an equilateral irregular hexagon
(FIG. 11c) with the angles 72.degree., 72.degree., 216.degree.,
72.degree., 72.degree., 216.degree. by the lines (a), (b), or (c),
or the corners of an equilateral irregular hexagon (FIG. 4) with
the angles 36.degree., 144.degree., 108.degree., 108.degree.,
72.degree., 252.degree. by the lines (a), (b) or (c), or the
corners of an equilaterally irregular octagon FIG. 5) with the
angles 108.degree., 36.degree., 252.degree., 72.degree.,
252.degree., 36.degree., 108.degree., 216.degree. by the lines (a),
(b) or (c), or the corners of an equilateral irregular octagon
(FIG. 6) with the angles 36.degree., 108.degree., 144.degree.,
252.degree., 26.degree., 108.degree., 144.degree., 252.degree. by
the lines (a), (b) or (c), wherein the connecting lines (a)
represent straight lines (b) represent centrally symmetrical,
non-straight lines (c) represent mirror symmetrical, non-straight
lines.
7. Structural elements, in particular puzzle element, tile or inlay
piece, characterized in that the structural element has a
circumference as it is shown in one of the Figures or respectively
such a circumference with rounded corners.
8. Group of congruent structural elements, characterized in that it
comprises a plurality of structural elements in accordance to one
of the claims directed to structural elements.
9. Group of structural elements in accordance with one of the
claims directed to groups of structural elements, characterized in
that the group comprises of one or more of the following features:
a) The structural elements have rounded corners; b) The structural
elements differ in their surface design; c) The structural elements
have approximately rectangular cross-section d) The subgroup UA has
at least three, preferably at least four structural elements with
congruent circumference; e) The thickness of the structural
elements is relatively small in comparison to the length of the
side of the GUF respectively GUZ respectively GUS; f) The
structural elements arc tiles, inlay elements or placement game
elements, in particular puzzle elements; g) The corners of the
rectangular irregular polygons are connected by rounded point
symmetrical lines, preferably forming structural elements with back
cuts to hook the structural elements to each other.
10. Use of the structural elements and group of structural elements
in accordance with one of the claims directed to such groups or
respectively structural elements in placement games, puzzles, inlay
work, jewelry, tile sets, fabrics, wallpapers or to cover surface
arcas, wall areas or furniture surfaces.
11. Placement game, in particular puzzle, characterized by at least
a set of structural elements in accordance with one of the
preceding claims and preferably by a closed edge area which
encompasses a placement area which can be filled by a number of
structural elements.
Description
[0001] The invention relates to structural elements as well as sets
of structural elements or respectively tiles with geometrically
defined circumferences which will be in the following also referred
to simply as "tiles", as well as to surface patterns composed of
such structural elements or respectively tiles. The invention also
relates to applications of these sets of structural elements in
tilings, puzzles, placement games, inlays, fabric patterns and
jewelry.
[0002] Several sets of tiles with geometrically defined
circumferences are known. Many of these sets of tiles can
tessellate the plane. Some more recent ones have been described in
Spektrum der Wissenschaft, January 2000, page 106 f. Such tile sets
cover the plane either periodically or non-periodically.
[0003] Several puzzles use tile sets with geometrically defined
structural elements, so the well-known Tangram or the pentagonal
puzzle of U.S. Pat. No. 4,343,471. These geometrical tile sets of
the puzzles comprise a number of different tiles.
[0004] The tile sets which cover the plane are generally limited
with respect to the possible structures. Thus tilings from square
or hexagonal tiles are only represented in a single periodic
structure.
[0005] For puzzles but also for tilings or tessellations such as
parquet or inlay structures tile sets would be of interest which
allow a variety of different structures or patterns in periodic or
non-periodical way, but are composed of only one or a very limited
number of different types of tiles.
[0006] In accordance with the invention a set of structural
elements or tiles with geometrical circumferences is provided as it
is defined in the claims. All tiles of these sets, which can of
course also be combined with other sets, consist of or derived from
equilateral, irregular pentagons with the internal angles
36.degree., 108.degree., 108.degree., 36.degree., 252.degree..
These equilateral pentagons will be referred to in the following
also as "Ipenta(s)". Sets of congruent Ipentas cover the plane in
accordance with the invention in many different ways, in particular
periodically, non-periodically, symmetrically and
non-symmetrically. The tilings can among other possibilities have a
decagonal base structure, a parallel gram structure as well as
various spiraling structures.
[0007] In U.S. Pat. No. 4,343,471 in FIG. 1 item "7" an Ipenta is
shown. This tile is a section of the well-known equilateral
pentagonal star cut along an internal diagonal the length of which
is equal to the length of the side of a star tip. A reference that
this tile can be used to cover the entire plane is absent in U.S.
Pat. No. 4,343,471.
[0008] In Scientific American 1977, page 110 ff. "Mathematical
Games" Martin Gardner describes various tilings, among them the
famous tiling by Roger Penrose. These tiles are not
equilateral.
[0009] In all plane covering tilings using the groups of Ipentas or
tiles derived from such Ipentas as described in the following,
always an even number of four to ten of such Ipentas meet each
other in each interstices or crossing point. Thus, each pattern can
be represented by tiles of two different colors in such a way, that
never two tiles of the same color will contact each other along a
side of the pentagon. Thus a quasi checkerboard structure
results.
[0010] The set or group of structural elements in accordance with
the invention can in particular also contain modifications of the
Ipentas as they will be described in more detail in the
following.
[0011] The sets of structural elements in accordance with the
invention can preferably be used in the following applications,
which themselves represent embodiments or respectively further
examples of the invention.
[0012] The sets of structural elements can be tile sets for the
inside or the outside. With these sets floor areas, places and
walls can be covered. Insofar these tiles can be made from
traditional materials such as ceramic, marble, precious stones,
plastic, metals or wood. The corners can be rounded, preferably
with equal radius of curvature in the comers or with curvature
sections of the smaller corner being repeated in the larger comers
in a convex or respectively concave manner three times in a
congruent fashion. Thereby the roundings in all patterns fit
smoothly together.
[0013] The structural elements can be used for placement games, in
particular for puzzles. These puzzles can be provided in a standard
fashion with pictures which are to be placed. These placements or
respectively puzzles in accordance with the invention preferably
have a solid and set frame area, or they can be cut straight in
order to provide e.g. a rectangular edge, whereby however edge
pieces are generated which differ from the congruent puzzle pieces.
In the embodiment of the placement game which has a solid fixed
frame with a non-smooth internal edge the structure elements can be
employed in a game in which two or more players alternate in
placing the structural elements and e.g. the player wins who places
the last structural element. Since the structural elements can be
placed in the range inside of the edge not only in a completely
space filling way, it is not a priori defined how many of such
structural elements can be placed. This depends rather exclusively
from how the structural elements are being placed. But alternating
placement of a structural element or also of a plurality of those
(e.g. determined by the number of a dice) interesting game
variations can result. The placement game in accordance with the
invention preferably have a larger number of structural elements,
in particular more than nine and the structural elements consist
all preferably in one or two sets of congruent structural elements
as well as optionally in the above mentioned frame area.
[0014] In many of the known puzzles none of the elements
geometrically fit onto the other, unless also the picture area fits
together. In accordance with the puzzle of the invention all pieces
match with all sides of the other pieces without the requirement
that the picture areas also fit. This increases the difficulty of
the puzzle.
[0015] In a particular embodiment of the puzzles or respectively
placement games of the invention, the sides of the Ipentas or
respectively the structures resulting from these Ipentas have been
replaced by line sections which upon placement of the puzzle pieces
will result in their hookup. For the Ipentas this can occur by
replacing each of the five sides by one bulging line which bulges
to the left and centrally symmetrically thereto to the right,
whereby back cuts or respectively hookups become possible.
[0016] In accordance with the invention, the sets of the structural
elements can be made from wood, stone, ceramic, glass, plastic,
metal or precious stones and be formed into inlays. These
structures can, in spite of their geometrical rigidity, formed into
a variety of pictures by multiple kinds of positioning of the
structural elements.
[0017] Most arrangements of the Ipentas can be described by a
non-intersecting line sequence. When this line sequence is rounded
out, the contact areas disappear and a continuous, generally singly
connecting area results which can be for instance cut from veneer,
in particular by means of a laser, and can be placed into a plate
or section with the corresponding cutout. Such items, in particular
on surfaces of furniture, are a further embodiment of the
invention.
[0018] The set of structural elements can be composed into fabric
patterns or respectively wallpaper patterns or similar items, in
which the entire area is completely or substantially completely
covered by Ipentas or respectively modified Ipentas, wherein the
circumference however due to a randomization or respectively
accidental distribution do not repeat. In this case, the preferred
embodiment is made from a periodically repeating hexagonal
subdivision of eight Ipentas, in which the Ipentas, however, can be
statistically distributed differently from sub area to sub area, so
that in spite of the periodicity of the edge of the sub group a
statistically non-periodic arrangement of Ipentas results. For most
observers the hexagonal structure of the sub areas thereby
disappears.
[0019] A further embodiment of the invention are new structural
elements or respectively tiles which are derived from the Ipentas.
A first group of these structural elements results from the Ipentas
by replacing each side of the Ipentas by a line track, a vector
sequence or a curved section (in the following "curved section"),
in which the distance between the end points is the same and for
which at least one of these is non-straight. Preferably all five
curved sections are equal, resulting in either one or two new
modified Ipentas, which have five "main corners" which spread or
define an Ipenta. If the curved sections are centrally symmetrical,
then only one new structural element results. Otherwise two of
these arise, depending upon whether the replacement of the five
sides of the Ipenta by curve sections is done clockwise or
counterclockwise.
[0020] Among the new structural elements those are particularly
preferred, in which the curved sections have a centrally
symmetrical S- or respectively Z- shape. Among these again those
are particularly attractive for applications such as puzzles or
other placement games, in which the curve sections cause a hooking
or interlocking of the juxtaposed structural elements. The latter
can be achieved by providing that at least two partial sections of
the curve sections form an angle of more than 180.degree. with each
other. Preferably the curved sections neither intersect with
themselves nor do they intersect the curved sections of the other
sides of the Ipentas.
[0021] If the curved sections consist of two line elements forming
an angle of 144.degree. then two inventive structural elements with
eight (correct: ten) or respectively six corners arise. For the
structural elements with six comers one simply leaves out the
resulting double lines running back and forth, since these cover
zero area. In a more general sense the following can be stated: If
the line segment consists of a number of n>2 sections and the
angle between the first and the last section is 144.degree., then
the tessellation of Ipentas results in a tessellation made from two
groups of congruent tiles having a different circumference length
of the two tiles. The difference of the circumference length is
four times the overlap of the first and last section of the line
sections. Again one leaves out the overlapping line sections of the
smaller structural element.
[0022] In the case of the structural elements which have five
identical but not centrally symmetrical curved sections, two
different structural elements result (e.g. the just described
structural elements with ten (erroneously: Eight) or respectively
six corners), wherein each tiling or complete covering of the plane
or respectively with a plane section with these structural elements
comprises approximately the same number of the two different
structural elements.
[0023] The invention further provides new structural elements or
respectively tiles which are also derived from Ipentas and have a
circumference which results from placing at least two Ipentas along
one or two sides juxtaposed with each other. Preferred are the
structural elements from two so juxtaposidly positioned Ipentas,
whereby seven different (not counting mirror reversing) structural
elements result, six of those having eight and one of those having
six sides of equal length. Three of the structural elements can
tile the plane periodically or non-periodically, wherein only the
line symmetrical structural element allows the tiling with this
single structural element. The others form structural elements
which can tile the claim only together with the structural element
which is mirror symmetrical to them. These three structural
elements with eight and one with six sides are the preferred ones
in this group. Also with these structural elements each side can be
replaced by a curved section. Thereby again one or two groups of
structural elements result which can in the case of a tessellation
or tiling e.g. be interlocked with each other.
The invention will be described in the following in conjunction
with the drawing illustrating embodiments.
[0024] FIG. 1 to 3 shows different tesellations made from the sets
of Ipentas in accordance with the invention,
[0025] FIGS. 4 to 6 show tessellations by sets of modified
Ipentas,
[0026] FIG. 7a to 8h-2 show structural elements in accordance with
the invention from modified Ipentas,
[0027] FIG. 9a to g show structural elements in accordance with the
invention which are composed of two Ipentas,
[0028] FIG. 9h shows the structural elements of FIG. 9a-g put
together,
[0029] FIG. 10a to b shows structural elements in accordance with
the invention composed into a spirally tessellation,
[0030] FIG. 11a shows a tessellation from a group of structural
elements in accordance with the invention,
[0031] FIG. 11b to c shows two complementary structural elements in
accordance with the invention,
[0032] FIG. 11d shows a tessellation of structural elements in
accordance with FIG. 11b and 11c which is derived from the one in
accordance with FIG. 11a.
[0033] In FIG. 1 a non-periodic tessellation of Ipentas as well as
a cross-section along line A-A is shown. These Ipentas 1 and 2 all
congruent and have one or two colors. Two "circles" (decagons) from
ten Ipentas are shown in FIG. 1. The possibility to continue this
structure into infinity can be recognized.
[0034] In FIG. 2 a non-periodic, centrally symmetrical tessellation
from Ipentas 1 and 2 is shown. There, too, the neighboring
structural elements switch color.
[0035] In FIG. 3 a non-periodic tessellation from Ipentas is shown.
Each time eight of these Ipentas (four light ones and four dark
ones) form a hexagon. The circumference of these hexagons tile the
plane periodically. The Ipentas themselves however tile the plane
non-periodically, wherein the two possible arrangements of the
eight pentagons in the hexagon is determined by an arbitrary or
statistical distribution. However, also an algorithm can be used
for this.
[0036] In FIGS. 7a to 7h structural elements are shown which have
been derived from Ipentas. For these the sides of the Ipentas have
been replaced by centrally symmetrical line sequences appearing
edgy or rounded. This results in congruent structural elements for
the tessellations. Indentations which result in a secure connection
or hookup of the structural elements are shown in some of FIGS. 7a
to h.
[0037] In FIG. 8a-1,2 to 8h-1,2 respectively two structural
elements or respectively tiles are shown, which follow from the
Ipentas as follows: First each line of the Ipentas is replaced by
two lines of equal length having an angle of 144.degree. between
them. Thereby, as described above, two structural elements 9 and 10
as they are shown in FIGS. 11b and c in a tessellation in FIG. 11d,
result. If one now replaces each of these ten or respectively six
sides of the structural elements by the corresponding line
sequences or line shapes, as this has been also done with the
Ipentas themselves in the corresponding representations of FIG.
7a-h, then the shown modified structural elements result. The
structural elements 8a-1 and 8a-2 (and correspondingly for 8b-h)
tile together the plane, partially with interlocking or hookup.
[0038] In FIG. 9a-g seven structural elements are shown which
result from juxtaposing two Ipentas along one side or two sides. In
FIG. 9h it is shown how these seven structural elements can be put
into a shape of minimal circumference. From these structural
elements in accordance with the invention three structural elements
are particular preferred, because these either alone or together
with others or respectively with those appearing mirror symmetrical
to them the plane can be tiled periodically and non-periodically.
These are the structural elements in accordance with FIG. 9a, 9f,
9g. Thereby these, too, can be used for the above mentioned
applications. In the case of the structural element in accordance
with FIG. 9g this is an equilateral irregular hexagon which is
derived from two Ipentas, which are juxtaposed along two lines. In
the case of the structural elements of FIGS. 9a and 9f, these are
also equilateral, irregular octagonal structural elements which are
obtained by juxtaposition of two Ipentas along one line.
[0039] Non-periodical tessellations from the structural elements in
accordance with FIGS. 9a, 9f, 9g are shown in FIGS. 5, 6 and 4.
Only the structural element in accordance with FIG. 9a tiles or
tessillates the plane by itself, see FIG. 5. In the case of
structural elements in accordance with FIG. 9f and 9g one needs for
this purpose in addition the mirror symmetrical structural elements
as shown in FIG. 6 by the gaps between the structural elements (or
by white elements).
[0040] In FIG. 10a and b two tessellations with modified Ipentas 7
and 8 are shown which have a spiraling structure, and wherein FIG.
10b encompasses the arrangement of FIG. 10a and shows the growing
spiral. The individual structural elements 7 and 8 correspond to
the structural element shown in FIG. 7d.
[0041] In FIG. 11a-d it is shown how new structural elements 9 and
10 result from the Ipentas. The original tessellation from pure
Ipentas is shown in FIG. 11a. If one replaces in this starting
tessellation each side by a line sequence of two lines forming an
angle of 144.degree., then the tessellation in accordance with
Figure 11d results, wherein the double lines, as described, have
been omitted. In this way two sets of tiles which are individually
shown in FIG. 11b and FIG. 11c result. The first set consists of
equilateral decagonal structure elements, while the second set
consists of equilateral hexagonal structural elements. The
decagonal structural element is equilateral and has the angles
288.degree., 144.degree., 72.degree., 144.degree., 144.degree.,
144.degree., 144.degree., 144.degree., 72.degree., 144.degree.. The
hexagonal structural element has the same side length as the
decagonal structural element and the angles are 72.degree.,
72.degree., 216.degree., 72.degree., 72.degree., 216.degree..
[0042] Placement games based on the above tessellation preferably
in addition have a surrounding edge which does not touch the
tessellation, e.g. a rectangular edge, which confines the placement
game, which can for instance be cut from cardboard, plastic
material or wood, and thus defines a closed edge area which
surrounds the internal area. The internal area is filled by the set
of structural elements in accordance with the invention or
respectively by the two represented sets. Also placement games can
be formed in such a way that one side can be used as a puzzle and
the other side with no picture and with the structural elements of
the same color or with the structural elements of different colors,
e.g. up to six colors, can be formed. With this other side or the
so formed single sided placement games it is possible to play in a
multitude of ways games, in which one defines the winner as the one
who places the last element or one can use a dice to determine how
many elements a player has to place and the winner or loser is
defined by the number of placed structural elements or the number
of no longer placable structure elements. One can also establish
game rules in accordance with which it is the goal to connect with
one contiguous chain of ones own elements to edges, or to surround
areas defined by one color and thereby make the surrounded areas as
large as possible, wherein their size is defined by the number of
structural elements playable side by side into the surrounded area.
Thereby games result which are related to NIM-games or to the game
of GO, which however in addition addresses the understanding of the
player for the geometry of the structural elements and thus has a
further educating effect.
[0043] In a further embodiment the placement game in accordance
with the invention, which is particularly useful on trips, the
structural elements are limited by magnetic forces in their
movement. Also the structural elements with back cuts which define
the juxtaposition and ensure against shifts are useful for this
purpose.
* * * * *