U.S. patent application number 13/997249 was filed with the patent office on 2013-11-07 for geometrical shape apparatus.
The applicant listed for this patent is Stephen William Cruwys Brazier. Invention is credited to Stephen William Cruwys Brazier.
Application Number | 20130295548 13/997249 |
Document ID | / |
Family ID | 43599011 |
Filed Date | 2013-11-07 |
United States Patent
Application |
20130295548 |
Kind Code |
A1 |
Brazier; Stephen William
Cruwys |
November 7, 2013 |
GEOMETRICAL SHAPE APPARATUS
Abstract
Apparatus comprising a set of shapes, the shapes derived from a
hexagonal footprint, and wherein for each of the shapes, at least
some of the vertices of the hexagon joined with straight lines to
define a perimeter of a respective shape of the set.
Inventors: |
Brazier; Stephen William
Cruwys; (Eastleigh, GB) |
|
Applicant: |
Name |
City |
State |
Country |
Type |
Brazier; Stephen William Cruwys |
Eastleigh |
|
GB |
|
|
Family ID: |
43599011 |
Appl. No.: |
13/997249 |
Filed: |
December 22, 2011 |
PCT Filed: |
December 22, 2011 |
PCT NO: |
PCT/GB2011/052574 |
371 Date: |
June 22, 2013 |
Current U.S.
Class: |
434/365 ;
273/148R; 273/153R |
Current CPC
Class: |
G09B 19/025 20130101;
G09B 19/00 20130101; A63F 11/00 20130101; A63F 2009/068 20130101;
A63F 9/10 20130101; A63F 2009/0697 20130101; G09B 7/00 20130101;
A63F 9/0669 20130101; A63F 2009/0695 20130101; A63F 9/0612
20130101 |
Class at
Publication: |
434/365 ;
273/148.R; 273/153.R |
International
Class: |
A63F 11/00 20060101
A63F011/00; G09B 19/00 20060101 G09B019/00; A63F 9/10 20060101
A63F009/10 |
Foreign Application Data
Date |
Code |
Application Number |
Dec 24, 2010 |
GB |
1021988.9 |
Claims
1. Apparatus comprising a set of shapes, the shapes derived from a
hexagonal footprint, and wherein for each of the shapes, at least
some of the vertices of the hexagon joined with straight lines to
define a perimeter of a respective shape of the set, wherein the
plurality of shapes have side lengths of either x, 2x, or 3x,
wherein a shape that includes a side length of x or 2x
interfaceable with any other shape including a side length of x or
2x; and a shape including a side length of 3x interfaceable with a
shape including a side length of 3x, and all of the shapes include
a side length of x, and a first subset of the shapes include a side
length of 2x, and a second subset of the shapes includes a side
length of 3x, where x is the side length of the hexagonal
footprint.
2. The apparatus of claim 1 wherein for at least some of the shapes
the centre point of the hexagonal footprint is joined to two
vertices.
3. The apparatus of claim 1, wherein each shape is derived by
selecting a starting point at a vertex or centre point of the
hexagonal footprint; joining the starting point, in sequence, to
one or more further vertices or the centre point; and returning to
the starting point, where the sequence defines the shape.
4. The apparatus of claim 1 comprising eighteen shapes; wherein
there are nineteen shapes when a hexagon shape is included.
5. The apparatus of claim 4 wherein five of the nineteen shapes are
reflectively asymmetric, and wherein five further shapes are
defined by reflections of the five reflectively asymmetric shapes,
to provide a set of twenty four shapes within the hexagonal
footprint.
6. The apparatus of claim 5 wherein the total area of the twenty
four defined shapes is 78y, where y is the area of one of six
equilateral triangles configurable to form the hexagonal
footprint.
7. The apparatus of claim 1 wherein the footprint and the shapes
are defined in two dimensions.
8. (canceled)
9. The apparatus of claim 1 further comprising a bounded area of
particular outline arranged to receive the shapes.
10. The apparatus of claim 9 further comprising a shape placement
region arranged to permit deployment of all shapes at the same
time.
11. The apparatus of claim 9, wherein the shape placement region
has an area of at least 72y, wherein y is the area of one of six
equilateral triangles configurable to form the hexagonal
footprint.
12. The apparatus of claim 11, wherein the shape placement region
has an area of at least 78y.
13. The apparatus of claim 9, wherein the shape placement region is
of substantially hexagonal shape.
14. The apparatus of claim 13 wherein the shape placement region
has a hexagonal footprint having an area of 96y or greater.
15. The apparatus of claim 9, wherein there are a plurality of
configurations of the shapes which can be accommodated by the shape
placement region.
16. The apparatus of claim 1 further comprising a number of
additional shapes, being duplicate shapes of one or more of the set
of shapes.
17. The apparatus of claim 16, wherein a composite shape is capable
of being formed using each of the set of shapes and one or more of
the duplicate shapes.
18. The apparatus of claim 4, further comprising a subset of the
set of the nineteen shapes, arranged to allow the selection of the
subset of shapes to construct a hexagonal shape within a hexagonal
outline of a shape placement region having a substantially
hexagonal shape.
19. The apparatus of claim 1 further comprising instructions
relating to use of the shapes as at least one of a game, a puzzle,
a gaming apparatus and an educational apparatus.
20. The apparatus of claim 1, wherein the apparatus is arranged to
allow a user to configure the relative positions of the shapes.
21. The apparatus of claim 1, further comprising a shape placement
template for use with the shapes which comprises a bounded area of
particular outline arranged to receive the shapes.
22. A The apparatus of claim 21, wherein the shape placement
template comprises multiple shape placement sub-regions.
23. Computer-readable instructions for a data processor, which,
when executed by the data processor, cause a visual display
apparatus to display a set of shapes, the shapes derived from a
hexagonal footprint, and wherein for each of the shapes, at least
some of the vertices of the hexagon joined with straight lines to
define a perimeter of a shape.
24. Apparatus comprising a data processor, a visual display device,
and the data processor configured to cause the visual display
device to display a set of shapes, the shapes derived from a
hexagonal footprint, and for each of the shapes, at least some of
the vertices of the hexagon joined with straight lines to define a
perimeter of a shape.
25. The apparatus of claim 24, wherein the apparatus is configured
to allow a user to provide an input to allow a user to determine
relative displayed positions of at least some of the shapes.
26. The apparatus of claim 24, wherein the data processor is
configured to cause the visual display device to display a shape
placement region into which a user may cause the shapes to be
located.
Description
TECHNICAL FIELD
[0001] The present invention relates generally to geometrical shape
apparatus.
BACKGROUND
[0002] We have realised that geometrical shape apparatus which
includes a set of shapes derived from a hexagonal footprint can
advantageously provide numerous applications and activities.
SUMMARY
[0003] According to a first aspect of the invention there is
provided apparatus comprising a set of shapes, the shapes derived
from a hexagonal footprint, and for each of the shapes, at least
some of the vertices of the hexagon, preferably including the
centre point, joined with straight lines to define a perimeter of a
respective shape of the set.
[0004] Various embodiments of the invention seek to provide a
comprehensive and versatile system in which the combination of
two-dimensional, inter-related, geometrical shapes give rise to a
variety of activities and applications including puzzles; pattern
making apparatus; physical games--board games, playing card games,
and dice games; digital applications; educational resources;
aptitude and intelligence tests; mathematical based problem
solving; and general recreational activities.
[0005] The shapes are preferably user configurable to allow sides
of the shapes to combine/interface, or at least be arranged in
registration with each other. In one embodiment of the apparatus
there is provided a set of shape pieces having sides which are
interfaceable with each other, and which pieces are individually
moveable/positionable.
[0006] One embodiment of the invention comprises a shape placement
region in the form of a template, onto which a user is required to
place shapes to fill the shape placement region. Advantageously, we
have realised that the shapes can be arranged to form
configurations of numerous symmetrical outlines, and so the shape
placement region may take the outline of any of those symmetrical
outlines.
[0007] According to a second aspect of the invention there are
provided instructions for a data processor, which, when executed by
the data processor, cause a visual display apparatus to display a
set of shapes, the shapes derived from a hexagonal footprint, and
wherein for each of the shapes, at least some of the vertices of
the hexagon, preferably including the centre point, joined with
straight lines to define a perimeter of a respective shape of the
set.
[0008] The instructions may be recorded on a data carrier, may be
in the form of a signal, or may be in the form of a software
product or other computer readable medium.
[0009] According to third aspect of the invention there is provided
apparatus comprising a data processor, a visual display device, and
the data processor configured to cause the visual display device to
display a set of shapes, the shapes derived from a hexagonal
footprint, and wherein for each of the shapes, at least some of the
vertices of the hexagon, preferably including the centre point,
joined with straight lines to define a perimeter of a respective
shape of the set.
[0010] Preferably the apparatus configured to allow a user to
provide an input to allow a user to determine relative displayed
positions of at least some of the shapes. The display device
preferably displays a graphic user interface.
[0011] It will be appreciated that not all shapes need to be
displayed simultaneously.
BRIEF DESCRIPTION OF THE DRAWINGS
[0012] Various embodiments of the invention will now be described,
by way of example only, with reference to the following drawings in
which:
[0013] FIG. 1 shows a basis for defining a set of polygons from a
hexagonal footprint,
[0014] FIG. 2 shows a set of shapes,
[0015] FIG. 3 shows a subset of the set of shapes of FIG. 2
together with the reflections of the subset of shapes,
[0016] FIG. 4 shows basis for defining the set of shapes,
[0017] FIG. 5 is a pictorial explanation of edge lengths of the
shape set of the shapes of FIG. 4,
[0018] FIG. 6 illustrates various ways in which some of the shapes
of FIG. 4 may be connected,
[0019] FIG. 7 illustrates area properties of some of the shapes of
FIG. 4,
[0020] FIG. 8a illustrates various hexagonally shaped surfaces,
[0021] FIG. 8b illustrates various hexagonally shaped surfaces,
[0022] FIGS. 9a to 9c illustrate the outline of shape placement
regions,
[0023] FIGS. 10a to 10q illustrate possible shape
configurations,
[0024] FIGS. 11a to 11f illustrate possible shape
configurations,
[0025] FIG. 12 illustrates additional repeated shapes,
[0026] FIGS. 13a to 13w show possible shape configurations,
[0027] FIG. 14 shows three possible shape configurations,
[0028] FIG. 15 shows a shape placement activity,
[0029] FIG. 16a illustrates a hexagonal footprint as a shape
placement region for a set of polygons,
[0030] FIG. 16b illustrates various orientations of shapes within
the shape placement region,
[0031] FIG. 17 illustrates various additional shapes,
[0032] FIGS. 18 and 19 illustrate shapes placed within the shape
placement region of FIG. 16a,
[0033] FIGS. 20a and 20b show two-player apparatus comprising a
shape set and a playing board,
[0034] FIGS. 21a and 21b show multiplayer apparatus comprising a
shape set and a playing board,
[0035] FIG. 22 shows various turns in use of either of the
apparatus of FIG. 21a or FIG. 21b,
[0036] FIGS. 23a and 23b shows a sequence of six images displayed
by a visual display device of an electronic application
apparatus,
[0037] FIG. 24 illustrates shapes corresponding to different levels
of difficulty of the electronic application apparatus of FIG.
23,
[0038] FIG. 25 shows a sequence of images of a game apparatus,
[0039] FIG. 26 shows various winning hands of playing card based
activities,
[0040] FIG. 27 illustrates a hierarchy of shapes,
[0041] FIG. 28 illustrates three dice,
[0042] FIG. 29 illustrates a playing card and the shapes required
on three throws of the dice of FIG. 28,
[0043] FIG. 30 shows the various areas of each of a set of
shapes,
[0044] FIGS. 31a to 31d show various embodiments of apparatus
comprising a plurality of shapes,
[0045] FIG. 32 shows shape placement templates for activities
relating to FIGS. 10a to 10q,
[0046] FIG. 33 shows shape placement templates for activities
relating to FIGS. 10a to 10n and 10q,
[0047] FIG. 34a shows a shape placement template,
[0048] FIG. 34b illustrates a possible shape configuration, and
[0049] FIGS. 34c and 34d illustrate possible design attributes for
the shape configuration.
DETAILED DESCRIPTION
[0050] Apparatus will now be described which comprises a finite set
of polygons that are derived from the footprint of a regular
hexagon by joining some or all of the vertices, including the
centre point, with straight lines, as shown in FIG. 1.
[0051] Eighteen different polygons can be created from this
principle; some familiar `regular` shapes; other shapes are less
obvious and mostly `irregular`. All shapes are contained within the
footprint of a regular hexagon. For some applications, the set
includes nineteen shapes with the inclusion of the hexagon shape
itself, as shown in FIG. 2.
[0052] Fourteen out of the eighteen polygons have reflective
symmetry of at least order 1. The remaining five shapes do not
possess line symmetry, therefore it can be argued that each of
those five shapes can be mirrored by reflection to create an
additional five shapes. In that regard, reference is made to FIG.
3.
[0053] Therefore, the set comprises twenty four `hexagonally based`
polygons when the original hexagon is included.
[0054] Each polygon is defined within the area or `footprint` of a
regular hexagon and each straight edge of every resulting polygon
is limited to lines that join various combinations of the six
vertices and centre point together, as best seen in FIG. 4.
[0055] For each polygon only three possible `edge` lengths occur. A
regular hexagon has an area equivalent to the combination of six
equilateral triangles placed together. Assuming that the length of
each side or `edge` of the equilateral triangle is one unit, then
the perimeter of the hexagon will be six units. Therefore, all
edges of each polygon will be either A, one unit length; B, two
unit lengths; or C, 3; the line that bisects (cuts in half) two
equilateral triangles. (Using Pythagoras, 2 squared-1 squared=the
square root of 3.apprxeq.1.73). In this regard, reference is made
to FIG. 5.
[0056] Sides A and B are related in length as B is twice the length
of A. Shapes that possess sides containing one or more sides of
length A and B will connect with each other in the configuration
A-A, A-B, B-B. If a shape contains a side of length C it may only
connect with another shape containing at least one side of length
C, ie C-C, as seen in FIG. 6.
[0057] All subsequent activities and applications of the set
require the connecting together of some or all of the set of twenty
four two-dimensional geometrical polygons in which the properties
of each shape, and particularly the `area`, are significant
factors. As the (shape-originating) original hexagon is made up of
six equilateral triangles, it is assumed that each equilateral
triangle has an area of one square unit (1y). If one equilateral
triangle is bisected into two, the resulting shape could be
connected to form an isosceles triangle, also of area one sq. unit.
These triangles represent two out of twenty four of the original
polygons of the set and because they both have an area of one sq.
unit, and are helpful in calculating and understanding the `area`
properties of all other members of the set. Reference is made to
FIG. 7.
[0058] The total area in square units of the twenty four polygons
in the set is seventy eight sq. units (78y). A guide to all
internal areas of the polygons contained in the set is shown in
FIG. 30.
[0059] A starting point in developing applications and activities
that involve the set is to explore potential designs for a surface
(board) on which to place the shapes. To use at least all of the
twenty four shapes once without leaving gaps (assuming that they
can be configured to combine appropriately), requires a board of
area 78 sq. units. To use a hexagonally shaped surface, the area
increases as follows 6, 24, 54, 96, 150, etc. Reference is made to
FIG. 8a.
[0060] The twenty four shapes, with an area of 78 sq. units will be
too large for the board of 54 and leave gaps or spaces on a board
of 96.
[0061] However, within these parameters, it is possible to create
two additional hexagons, one of area 18 sq. units and the other, 72
sq. units. Reference is made to FIG. 8b.
[0062] To provide flexibility when configuring shapes together on a
playing surface or `board` the design selected in FIG. 9a is
preferred. The total unit area available in this shape is 114 sq.
units. Other playing surface options are shown in FIG. 9b and FIG.
9c.
[0063] The hexagon shape of area 72 sq. units suggests in theory
that the set that comprises twenty four shapes with a total area of
78, less the original hexagon shape itself of area 6 sq. units,
could be configured to complete the hexagon pattern of area 72 sq.
units. The theoretical possibility is realised and the solution
outlined in FIG. 10a. This result is highly significant as it
demonstrates that a larger hexagon of area 72 has been configured
from the 23 unique shapes that derive from the original smaller
hexagon (the twenty-fourth shape).
[0064] For reference purposes this `solution` is referenced as
Puzzle Solution 1 (PS1). All subsequent solutions and patterns that
are deemed significant are similarly labelled.
[0065] PS2, shown in FIG. 10b, is a 24 sided polygon that resembles
a `star`. It comprises two hexagrams, one large, one smaller,
overlapping each other about a common centre and displaced by
30.degree..
[0066] PS3, shown in FIG. 10c, is a 36 sided polygon that comprises
6 clusters of 3 pointed vertices. All outside edges are the same
unit length.
[0067] PS4, shown in FIG. 10d, is a 30 sided polygon that resembles
a regular hexagon with 6 clusters of 4 units removed from each
`corner`.
[0068] PS5, shown in FIG. 10e, is a 24 sided polygon that resembles
a regular hexagon with 6 clusters of 3 units removed from each
`side` and a unit `isosceles` triangle removed from each of 6
`corners`.
[0069] PS6, shown in FIG. 10f, is a 24 sided polygon that resembles
a regular hexagon with 6 clusters of 4 units removed from each
`side`.
[0070] PS7, shown in FIG. 10g, is a 30 sided shape with a hexagonal
shape missing from its centre. It resembles a regular hexagon with
6 clusters of 3 units removed from each `side`. All outer edges are
the same length.
[0071] PS8, shown in FIG. 10h, is a 30-sided shape with a hexagonal
shape missing from its centre. It resembles a regular hexagon with
6 clusters of 2 units removed from each `side` and a unit
`isosceles` triangle removed from each of 6 `corners`.
[0072] PS9, shown in FIG. 10i, is an 18 sided shape that resembles
a regular hexagon with 6 clusters of 2 units removed from each
`corner` and a `hole` in the middle in the shape of a hexagram.
[0073] PS10, shown in FIG. 10j, is a 36 sided shape that resembles
a regular hexagon with 6 clusters of 2 units removed from each
`side` and a `hole` in the middle in the shape of a hexagram.
[0074] PS11, shown in FIG. 10k, is a 6 sided shape that resembles a
regular hexagon with a `hole` in the middle, also in the shape of a
hexagon.
[0075] Solutions demonstrated in FIGS. 10a-10q require all 23
original shapes and omit the original hexagon shape. It is likely
that PS1-PS11 (FIGS. 10a-10k) represent the only solutions in which
the 23 shapes of area 72 are configured together to create outcomes
that posses both line and rotational symmetry of order 6. All of
these solutions resemble modifications to a regular hexagonal
form.
[0076] In four of the six remaining solutions in this category,
PS12-14, FIGS. 10l-10n, each possess three lines of symmetry and
rotational symmetry of order 3 and PS15, FIG. 10o, demonstrates two
lines of symmetry and rotational symmetry of order 2, and is
possibly unique in this attribute.
[0077] PS16, FIG. 10p, also exhibits three lines of symmetry and
rotational symmetry of order 3. However, an outline template for
PS14 requires a different shaped surface from that provided for the
other solutions in this category. See FIG. 32p.
[0078] PS17, FIG. 10q, represents one example only of a solution
that exhibits rotational symmetry of order 6 with no line symmetry.
It is possible to create additional shapes that have this property.
It is likely that other shapes in this category that possess an
area of 72 sq. units using the 23 original shapes that possess
three lines of symmetry and/or rotational symmetry of order 3 can
be created.
[0079] Shape placement templates for PS1-PS17 are shown in FIGS.
32a to 32q and a different shape placement template for PS1-14, 17,
is provided in FIGS. 33a to 33o. Each template comprises a bounded
area in which the shapes can be placed.
[0080] Five other solutions that use all 24 shapes (including the
original hexagon shape), are described in FIGS. 11a-11e.
[0081] PS18, shown in FIG. 11a, is similar to PS7 however in this
example, it is a polygon as the centre is solid. All other
properties are the same as PS7.
[0082] PS18 represents the first, and so far only solution that
combines all 24 shapes in a `polygon` that possesses both
reflective and rotational symmetry of order 6. The solution is
particularly elegant as the hexagon shape is configured in the
center.
[0083] PS19, shown in FIG. 11b, is similar to PS11 with the
exception that the hexagonal `hole` in the middle is smaller in
area and displaced by 30.degree. about the centre point.
[0084] PS20, shown in FIG. 11c, is similar to PS9 with the
exception that the missing `hole` in the middle is in the shape of
a hexagon. All other properties are as PS9. Although not a true
`polygon` (as the middle is missing), PS20 also represents a
solution that contains all 24 shapes together in a symmetrical
(reflective as well as rotational) configuration of order 6.
[0085] PS21, shown in FIG. 11d, is a `polygon` that derives from
combining all 24 shapes together in a configuration that contains
both reflective and rotational symmetry of order 3.
[0086] PS22, shown in FIG. 11e, is also a `polygon` that derives
from combining all 24 shapes together in a configuration that
contains both reflective and rotational symmetry of order 2 and
gives the appearance of thirteen joined hexagons in rows of four,
five and four.
[0087] PS23, shown in FIG. 11f, has a hexagonal shaped hole in the
middle and contains both reflective and rotational symmetry of
order 2.
[0088] The combination of shapes used in creating each of the
patterns portrayed in PS1-23 provide one of many possible
configurations of fitting together the 23 or 24 shapes to achieve
the overall solution. It is not known exactly how many different
combinations will result in each pattern.
[0089] Configurations demonstrated in PS1-PS23 that use either 23
or 24 shapes of the `set` and exhibit both line symmetry and/or
rotational symmetry are referred to as `special` solutions. (NB
Solutions that have any shape of `hole` missing from within the
shape are not polygons).
[0090] A defined number of additional `duplicate` shapes are
introduced in order to increase accessibility, provide greater
flexibility of possible solutions, and introduce a wider range of
levels of difficulty. All four types of additional shapes are
duplicate shapes that derive from the original twenty four shape
`set`, as shown in FIG. 12.
[0091] The twenty additional shapes provide an additional area of
30 sq. units. The total area of a configuration using all shapes
would be 78+30=108. The enlarged set is referred to as the enlarged
set.
[0092] Applications and activities that require the use of
additional shapes and aimed at experimenting and investigating the
properties and configurations of the shapes are not restricted to
the number of shapes used. However, for all outcomes labelled
`solutions`, all twenty four shapes of the set should be used at
least once. In addition, each solution should possess rotational
symmetry of a least order 2 and/or at least one line of reflective
symmetry. A solution will range in area from 78-108.
[0093] Examples referred to as PS24-PS46 demonstrate the rich
variety of potential composite configurations that are possible to
construct using the (basic) set and the enlarged set, as shown in
FIGS. 13a to 13w. It should be noted that there will be other
configurations of the shapes that generate the same result for each
puzzle and that the solutions shown above are not unique.
[0094] Composite `Solutions` include: an equilateral triangle of
100 sq. units; a hexagon of 96 sq. units; a hexagram (a star of
David) of 108 sq. units; an equilateral triangle of 108 sq. units;
rectangles of various dimensions, (NB A square is impossible); two
joined hexagons; two hexagons each of area 54 sq. units; four
hexagons each of area 24 sq. units; multi-hex; an `irregular`
trapezium; a hexagon with protruding regions; a `setup`
configuration of area 108 sq. units; a hexagram of area 102 sq.
units; a `circle`; a rhombus; a trapezium; a polygon with 36 edges;
a polygon with 48 edges; a polygon with 42 edges.
[0095] The (basic) set and enlarged set provide numerous potential
outcomes and possibilities. By experimenting with various
configurations of the shapes, different and unexpected results
emerge, as exemplified in FIG. 14.
[0096] It is intended that guidelines/instructions for using and
manipulating the shapes are provided to users along with outline
templates of various shape configurations in which the task is to
complete the inside of the pattern without leaving gaps while at
the same time trying to use the minimum number of shapes. See FIG.
15. Previous examples in FIGS. 10, 11 and 13 can be presented as
tasks or puzzles in outline template form (as described above) that
provide varying degrees of difficulty.
[0097] Single user activities using the physical pieces of the
shapes can be summarised as follows:
[0098] PA 1--Provide a shape description in words. The user then
has to re-create the shape. This format could be used as a basis
for a competition eg Create a hexagon that uses all 23 shapes of
the basic set but not the original hexagon piece.
[0099] PA 2--Provide an outline template with
guidelines/instructions. The user then has to fill in the outline
with a combination of the shapes, for example as shown in FIG.
15.
[0100] PA 3--Provide the enlarged set. The user experiments and
plays with various configurations of shapes as an extension to
traditional building bricks and mosaics.
[0101] Shape placement templates provide activities of varying
difficulty and may be catagorised as follows: [0102] 1. Templates
that require two or more shapes from the (basic) set of shapes to
be configured together without leaving any gaps within the
template. [0103] 2. Templates that require the (basic) set of
shapes as well as some or all pieces of the enlarged set. [0104] 3.
Templates that require the (basic) set of shapes less the
(shape-originating) original hexagon piece, encompassing an area of
72y. [0105] 4. Templates that require the (basic) set of shapes
including the (shape-originating) original hexagon piece,
encompassing an area of 78y.
[0106] Detailed properties for shape placement templates for
category 3 and 4, and pertaining to FIGS. 10 and 11 follow:
[0107] The shape placement region of 72y preferably has an outline
arranged to accommodate the shapes in at least one of: a 6 sided
polygon in the pattern of a regular hexagon encompassing an area of
72y; a 24 sided polygon in the pattern of a `star` with 12 vertices
comprising of two hexagrams, one large, one smaller, overlapping
each other about a common centre and displaced by 30.degree.
encompassing an area of 72y; a 36 sided polygon that comprises 6
clusters of 3 pointed vertices in which all outside edges are the
same unit length encompassing an area of 72y; a 30 sided polygon
that resembles a complete hexagon of 96y with 6 clusters of 4y
removed from each `corner` encompassing an area of 72y; a 24 sided
polygon that resembles a complete hexagon of 96y with 6 clusters of
3y removed from each `side` and 6 clusters of 1y removed from each
`corner` encompassing an area of 72y; a 24 sided polygon that
resembles a complete hexagon of 96y with six clusters of 4y removed
from each `side` encompassing an area of 72y; a 30 sided shape that
resembles a complete hexagon of 96y with 6 clusters of 3y removed
from each `side` and less the further accommodation of 6y at its
centre in the shape of a hexagon encompassing an area of 72y; a 30
sided shape that resembles a complete hexagon of 96y with 6
clusters of 2y removed from each `side` location, 6 clusters of 1y
removed from each `corner` location and less the further
accommodation of 6y at its centre in the shape of a hexagon
encompassing an area of 72y; an 18 sided shape that resembles a
complete hexagon of 96y with 6 clusters of 2y removed from each
`corner` location and less the further accommodation of 12y at its
centre in the shape of a hexagram encompassing an area of 72y; a 36
sided shape that resembles a complete hexagon of 96y with 6
clusters of 2y removed from each `side` location and less the
further accommodation of 12y at its centre in the shape of a
hexagram encompassing an area of 72y; a 6 sided shape that
resembles a regular hexagon of 96y less the accommodation of an
area of 24y in the shape of a hexagon at its centre encompassing an
area of 72y; a pattern in the shape of twelve joined hexagons in
rows of two, three, four and three, encompassing an area of 72y; a
pattern of a hexagon encompassing an area of 96y, less the
accommodation of an area of 3 clusters of 8y at intervals of
120.degree. encompassing an area of 72y; a pattern of a hexagon
encompassing an area of 96y, less the accommodation of an area of 6
clusters of 1y at each `corner` location, less the accommodation if
18y at its centre in the form of 3 clusters 6y encompassing an area
of 72y; a pattern of a rectangle encompassing an area of 72y; a
pattern that resembles a triangle less 1y at each of three
locations at the mid-points of the sides of the triangle
encompassing an area of 72y; and a pattern of a hexagon
encompassing an area of 96y, less the accommodation of 6 clusters
of 2y from each `corner` location and a further 6 clusters of 2y
from each adjacent `side` location encompassing an area of 72y.
[0108] The shape placement region of 78y preferably has an outline
arranged to accommodate the shapes in at least one of: a hexagon of
96y, less the accommodation of an area of 3y at each of six `side`
locations encompassing an area of 78y; a regular hexagon of 96y,
less the accommodation of an area of 18y at its centre in the shape
of a regular hexagon encompassing an area of 78y; a hexagon of 96y,
less the accommodation of an area of 6y at its centre and less the
further accommodation of 2y at each of six `corner` locations
encompassing an area of 78y; a pattern that resembles a triangle
with the addition of 1y at each of three locations at the
mid-points of the sides of the triangle encompassing an area of
78y; a pattern of thirteen joined hexagons in rows of four, five
and four encompassing an area of 78y and; a hexagon encompassing an
area of 96y, less the accommodation of an area of 6y at its centre
and less the further accommodation of 1y at each of six locations
and a further 3y at each of two locations encompassing an area of
78y.
[0109] The pattern of the shape placement region may possess line
and/or rotational symmetry.
[0110] A composite shape is preferably capable of being formed
using each of the set of shapes and one or more duplicate
shapes.
[0111] The composite shape preferably possesses rotational symmetry
of order of at least two and/or possesses at least one line of
reflective symmetry.
[0112] The basic set and enlarged set provide the apparatus to
generate various applications and activities. Some of these have
involved creating patterns within a hexagonal surface of area 24,
54, 72 and 96. Also, the shapes have been used to configure other
regular and irregular shapes. The next step is to create activities
and applications that are based on the smallest hexagon possible. A
hexagon of area 6 sq. units, as shown in FIG. 16.
[0113] The `unit` hexagon contains all the shapes possible that can
be generated to create the shape set.
[0114] Shapes can be orientated within the hexagonal footprint in 6
`rotated` positions. In the unique case of the unit `isosceles`
triangle, it can be located in 12 positions. See FIG. 16b.
[0115] Two methods can be applied to create applications from the
hexagonal footprint or shape placement region. The first method
involves `filling` or `completing` the hexagon. For example, the
basic set comprising twenty four unique shapes are placed in
container A, and the 20 `additional` shapes in container B
(Reference is made to FIG. 17).
[0116] By selecting a shape at random from A, the aim is then to
complete the hexagon by selecting the appropriate shape(s), also at
random, from container B. The example shows both a negative and a
positive outcome. Reference is made to FIG. 18.
[0117] The second method involves re-creating the shapes from the
basic set. A shape from container A is chosen at random and
participants have to re-create the shape with shapes from Container
B. Reference is made to FIG. 19. These themes and applications that
result are now explored in more detail.
[0118] In a two-player application, the objective is to remove all
shapes from the playing surface, with the hexagon the last
remaining shape. Only the original twenty four shapes are point
scoring shapes. Each shape has a points score according to the area
of the shape, as shown in FIG. 30. The twenty `additional` shapes
are divided equally between the two players and marked differently
so that each player can distinguish between scoring shapes;
opponent shapes; and their own shapes. Shapes are placed on the
playing surface according to either the diagram in FIG. 20a or FIG.
20b. Each player chooses a `scoring` shape that becomes the focus
their strategy and does not let other players know the shape they
have chosen. Each player takes it in turns to move a shape. Shapes
are positioned to be able to contain a scoring shape. A `scoring`
shape can only be removed and its points value obtained when it
fits exactly into the small hexagon without leaving a gap. A
player's `additional` shapes are also removed by the same method as
the `scoring` shapes, however, they have no points value, only
strategic value. Each time a shape is touched and moved constitutes
a `turn`. A player must remove all their additional shapes from the
main playing surface before the final hexagon shape is removed. A
player may remove an `additional` shape from the playing surface at
any time, however, this always constitutes a turn.
[0119] An extension to the game is for both players, having removed
all shapes including the final hexagon, to then replace them one by
one in turn. The points for any remaining shapes that won't fit are
subtracted from each players score. There are additional points
available in specific situations. If a player acquires two shapes
and one is the `reflection` of the other, an additional point is
scored. If a player acquires two shapes and together they fit
together to form a complete hexagon, an additional point is
scored.
[0120] In a multi-player game, the objective is to remove all
shapes from the playing area with the hexagon the last remaining
shape. The player with the highest score is the winner. Only the
original 24 shapes are point scoring shapes. Each shape has a
points' score according to the area of the shape (FIG. 30). All
shapes are placed on the playing surface according to the diagram
in either FIG. 21a or FIG. 21b.
[0121] In turn, each player removes one of the two triangular
`additional` shapes that have an area of 1 sq. unit. A player may
use a turn to exchange some or all of their accumulated
`additional` shapes for larger `additional` shapes if and when
required. For example, two equilateral triangles or two isosceles
triangles may be exchanged for a rhombus; a rhombus and an
equilateral triangle or two isosceles triangles and an equilateral
triangle may be exchanged for a trapezium and vice-versa. A player
aims to acquire sufficient `additional` shapes to re-create a
`scoring` shape that they have pre-identified without disclosing it
to the other players. If a player intends to attempt to `score`
they should declare a `play` at the beginning of their turn and
should remove the intended shape and place it in the centre of the
playing surface. Reference is made to FIG. 22.
[0122] In this example, Player A decides to accumulate three
`additional` shapes during three turns. At the beginning of turn 4
the player declares his intention to `score` by declaring a `play`
and removes the `scoring` shape they will attempt to re-create and
places it in the centre space. Player A then takes the three
`additional` shapes and positions them so that both the `scoring`
shape selected and the re-created shape are identical. If during
the `play` the attempt to re-create the shape fails, the player
returns the shape as well as all three additional shapes to the
playing surface. If the `play` is successful and the re-created
shape is identical to the `scoring` shape selected, the player
scores points according to the unit area of the shape (in this
example, 3 points as the shape has an area of three units), and the
shape is removed and kept by the player in order to accurately
total the points at the end of the game. In this example, Player A
could have decided that their intended `scoring` shape required
more or less "additional` shapes than three and therefore could
have declared a `scoring` move earlier or later. After a `play` has
been declared, whether successful or unsuccessful, all `additional`
shapes used in the `play` should be returned to play by replacing
them on the playing surface.
[0123] As with the two-user application, an extension to the game
is for all players, having removed all shapes including the final
hexagon, to then replace them one by one in turn. The points for
any remaining shapes that won't fit are subtracted from each
player's score. There are additional points available in specific
situations. If a player acquires two shapes and one is the
`reflection` of the other, an additional point is scored. If a
player acquires two shapes and together they fit together to form a
complete hexagon, an additional point is scored. When a player
declares a `play` it can be challenged by other players if they can
re-create the scoring shape with fewer shapes. In this scenario,
the challenging player would acquire the `scoring` shape and
allotted points. The player who declared the `play` unsuccessfully,
would forfeit the shapes they used in the `play` and return them
back to the playing surface.
[0124] Most, if not all, of the activities using the physical
shapes above, which could be made of any suitable material, can be
realised as digital applications. Applications (`Apps`), developed
digitally allow shapes to be manipulated using 1), handheld
devices--small displays such as iPods.RTM., iPhone.RTM., Nintendo
DS.RTM. etc, 2), computers and `tablet` devices--medium displays
such as iPads.RTM. and 3) interactive displays--large displays such
as interactive whiteboards and interactive horizontal table
devices. The most effective use of digital applications will be
using devices that incorporate `touch` technology. These devices
include a data processor, a visual display and instructions for the
data processor to generate a suitable display on the visual
display.
[0125] Electronic Application 1--This application requires
generation of a set of `digital` shapes that consists of the
extended set. The user has the ability to move the shapes into
various configurations. Two types of activity are offered. 1)
Challenges expressed in words such as, "Create a figure in the
shape of a hexagon that uses all twenty three shapes of the set but
not the original hexagon piece itself." 2) Outline templates that
require the user to use the minimum number of shapes necessary to
complete the inside of a template without leaving gaps. This can be
either timed or un-timed.
[0126] Electronic Application 2--This application requires a
playing surface of a single hexagon unit. The player starts with a
credit score of six points. By selecting Button A, one of the
twenty four shapes will appear, at random, within the footprint of
the hexagon. The potential score depends on the area of the shape
that appears. Should the hexagon shape itself appear, the player
automatically scores six points. However, for all other shapes, the
player should then select Button B to generate one of the
`additional` shapes which will appear to the side of the playing
surface. The `additional` shapes that appear should then be
manipulated into the footprint of the hexagon until eventually no
gaps are left. When the unit hexagon has been completed, the player
obtains the points of the `scoring` shape. If the `additional`
shape that appears does not `fit` into the footprint of the
hexagon, either the game is over--with no points, or, the player
can choose to continue but forfeits one point to discard the
`additional` shape each time a shape is generated that does not
fit.
[0127] Electronic Application 3--A digital version of two-user
application described above.
[0128] Electronic Application 4--A digital version of multi-user
application described above.
[0129] Electronic Application 5--This application requires a player
to complete the six hexagon cells located on the outside of the
playing surface by filling them with various shapes from the twenty
four shapes that comprise the basic set. At random, a shape appears
in the centre hexagon. It then begins to travel to the outside of
the playing surface following one of 6 six routes, as shown in FIG.
23. A player selects the shape, by tapping onto the visual display
screen, and manoeuvres it into position within one of the hexagon
cells in one continuous movement. If a shape is `tapped` and not
located into one of the outside cells within a
`reasonable`/predetermined amount of time, it will break free from
the `touch` and continue its movement to the outside of the playing
surface. Shapes may be moved in any direction and may also be
rotated. When a hexagon is complete, the contents of the hexagon
cell clears and six points are scored. If a shape reaches the
`outside` without being tapped or selected, points are lost
according to the value of the scoring shape. If an outside `cell`
is left partly filled for too long, it begins to pulsate. Unless
the cell is quickly completed, it will explode and the cell removed
leaving only 5 cells. If all cells on the outside are lost the game
is over. If a players score drops below zero the game is over.
[0130] The aim is to gain as many points as possible.
[0131] Shapes from the basic set are introduced to the game at
intervals that indicate eight levels of difficulty. Reference is
made to FIG. 24. Both `Scoring` shapes as well as `additional`
shapes appear in groups to ensure there exists a realistic
opportunity to be successful. Another method to increase the
complexity and difficulty of this application is to increase the
rapidity of emerging shapes from the centre and/or increase the
speed at which shapes move.
[0132] By the time Level 8 is achieved, any of the twenty four
`Scoring` shapes and all of the 20 `Additional` shapes will be in
play and may emerge from the hexagon at the centre.
[0133] Electronic Application 6--Reference is made to FIG. 25. This
application requires a player or players to place `tokens` within a
grid pattern. Tokens are then eliminated by movements within the
grid according to rules that govern the emerging set of shapes. The
player with the most tokens remaining wins. At the beginning of the
`play`, the nodes A-G light up in turn in a random order. The last
point to light up becomes the starting point of one round of the
game, in this example, E. Within the grid there are twenty four
sections for players to place their tokens. In this example, two
players each have six tokens each and place them in sectors
according to the diagram. Once tokens have been placed on the grid,
the game commences. A random path emerges starting from the node
labeled E and, following the straight lines of the grid, continues
until it reaches another node. On reaching another node, it changes
direction again and makes its way to the next. This continues until
the path returns to the node located at E. The only rules for the
construction of the path are: a) it does not cross an existing line
between nodes, b) it does not re-trace the path of an existing line
and c) it does not take a route that makes it impossible to return
to E without forming a polygon. In this way, the resulting
combination of straight lines will outline a polygon that
represents one of the twenty four shapes of the shape set. The
tokens that lie within the shape remain, those that do not are
removed. On random occasions, as the shape emerges it will either
start to rotate before finally settling into a final position from
which gains and losses can be assessed or the players can decide to
remove winning `tokens` before risking them to further movement of
the shape. Winners are players with remaining tokens or a winner is
the player with most remaining tokens. In this example Player 2
wins has he has one and only remaining token.
[0134] Many of the applications described above can be adapted as
playing card activities using four possible variants:
[0135] Type A `Scoring` cards are represented by one pack of twenty
four cards that represent the twenty four shapes.
[0136] Type B `Scoring` cards are represented by one pack of twenty
four cards that represent the twenty four shapes and a second pack
of cards that represent the `Additional` shapes and consist of
twenty cards.
[0137] Type C Both `Scoring` cards as well as `Additional` cards
are represented in one pack of forty four cards.
[0138] Type D `Additional` cards represented by a pack of twenty
cards.
[0139] When appropriate, more than one pack of each type of pack
may be used depending on the requirements of individual games.
[0140] Playing Card Application 1--Snap--Type A--All twenty four
`Scoring` cards are dealt to two players face down. Cards are
placed down in turn awaiting two similar shapes OR two shapes that
together make a hexagon. The first player to state `Snap` or
another pre-agreed word, wins the pile. The winner is the first to
gain all cards when the other player runs out of cards. Two packs
or more can be used for larger numbers of players.
[0141] Playing Card Application 2--Pairs--Type A--All twenty four
`Scoring` cards are placed individually face down in a rectangular
grid 6.times.4. Players take it in turn to try to gain a pair of
identical shapes by the turning over of two cards. If unsuccessful
the cards are replaced. If successful the cards are removed. The
winner is the player to gain most pairs. Two packs or more can be
used for larger numbers of players.
[0142] Playing Card Application 3--Black Jack or `21`--Type B--The
two sets of cards are set down in two separate stacks. Each player
receives one card from the `Scoring` cards pile. The aim is to add
cards from the second stack of `additional` cards to complete the
hexagon. The player may twist, stick or go bust in the same way
Black Jack or `21` is played with a traditional pack of playing
cards. Two packs or more can be used for larger numbers of
players.
[0143] Playing Card Application 4--Hex-it--Type B--The two sets of
cards are set down in two separate stacks. A card is turned face up
from the `Scoring` cards stack. The aim is to gather cards from the
second stack of `additional` cards to complete the hexagon. Players
take it in turn to add to their collection of cards up to a total
of six cards, keeping their cards hidden from their opponents. When
it is time to take a seventh card a player may either refuse to
take a card or take a card and replace one existing card back to
the bottom of the stack. When a player has the appropriate shapes
in their collection of cards to complete the hexagon they should
wait to the beginning of their next turn and declare `play` as
opposed to `pick-up`. If a player is successful and `plays` the
necessary shapes to complete the hexagon, they remove this card and
score points according to the unit area of the shape removed. They
also turn over the next card to instigate the next round of play.
If a player is unsuccessful, the card remains face up and play
continues. The unsuccessful attempt results in those accumulated
cards used in the play to be forfeited and be returned to the
stack. Two packs or more can be used for larger numbers of
players.
[0144] Playing Card Application 5--`Poker`--Type A--Two sets of
cards representing the `Scoring` cards only are set down. Each
player receives three cards which are kept hidden. Three cards from
the stack are then displayed. The three cards are added to one at a
time until six cards are displayed. A `round` can last from three
to six of the cards being displayed from the stack. Winning hands
and the hierarchy of shapes are defined in FIGS. 26 and 27. Two
packs or more can be used for larger numbers of players.
[0145] It will be appreciated that all descriptive names given
above to applications are working titles only and if they are
suggestive of regular playing card games it is because they possess
similar concepts.
[0146] The general rules are as follows: [0147] There are two
starting positions for this game. Both a five card version and a
six card version are viable. The six card version is described
above. The five card version means that each player receives two
cards instead of three and two cards are displayed and added to
until a maximum of five are displayed. A winning `Hex Run` in the
five card version will omit the `Hexagon` card. [0148] Same colour
combinations always takes precedence over two colours combinations
unless otherwise stated [0149] When a player receives the `Hex`
card or the `Hex` card comes into play, then making a complete
hexagon will triumph over all other hands except a `Hex Run`.
[0150] In the case of equivalent hands, card values are calculated
according to the Hierarchy of Shapes table, as shown in FIG.
27.
[0151] Physical Applications (as opposed to `digital` Applications)
that combine some of the mediums already described (such as shape
pieces, playing cards and dice) are proposed and outlined as
follows.
[0152] Combination Application 1--A game that uses the basic set of
twenty four `Scoring` shapes on playing cards while the four
`Additional` shapes are represented on three dice. Reference is
made to FIG. 28.
[0153] In this activity, a pack of cards containing the basic set
is placed face down. The top card is turned over. The player then
chooses one, two or three dice to attempt to `complete` the
hexagon. See example in FIG. 29.
[0154] An extension to this application is to use the dice only to
throw the shapes that will combine to make a hexagon.
[0155] Combination Application 2--This activity is similar to
Combination Application 1 except that a player throws between one
to three dice in order to re-create the shape that appears from the
pack of cards containing the set. The `joker` in the pack in this
game is the rectangle as it is impossible to re-create it with the
shapes on the dice. Therefore, if a player draws the `rectangle`
and the player is still in the game after a throw of three dice,
the player is offered an additional throw.
[0156] Combination Application 3--This activity requires a playing
surface and all shape pieces with two packs of cards (twenty four
original scoring shapes and twenty additional shapes).
[0157] A card is turned over face up from the top of the twenty
four original scoring shapes pack. This card represents the shape
that both players will now attempt to re-create by collecting turn
by turn, shapes from the second pack of twenty additional shapes.
The shape is identified on the board, removed and placed in the
centre of the playing surface to represent the focus of the
activity to follow. In turn, each player removes an `additional`
card until a player has sufficient `additional` shape cards to
re-create the `scoring` shape. If a player intends to attempt to
`score` they should declare a `play` at the beginning of their
turn. The player now declares the cards they intend to use to
re-create the shape in the centre by selecting the shapes from the
playing surface. The `additional` shapes are now maneuvered into
position to create a replica of the shape in the centre of the
board. If the attempt is successful, the player keeps the shape in
the center and scores the points according to its points value. The
cards the player has used to re-create the shape are returned to
the bottom of the pack. If the player is unsuccessful in their
attempt, the shape is returned to play and the player forfeits the
cards back to the pack.
[0158] Another embodiment of the invention comprises an `education`
pack which reflects the geometrical nature of the twenty four
shapes and the various configurations that are possible to create
as outcomes. It consists of individual, pair and group activities.
It provides activities and tasks that concentrate on analysis of
the geometrical properties of each shape. These include: [0159]
Properties of Polygons [0160] Special names--Triangles,
Quadrilaterals etc [0161] Special properties--Regular/Irregular;
Concave/Convex [0162] Angles--Acute, Obtuse, Reflex, Right-angled
[0163] Symmetry--Line, Point/Rotational [0164]
Measurement--Perimeter, Area [0165] Tessellations [0166]
Terminology
[0167] At a basic level, students are encouraged to `play` with the
shapes and to attempt to create both mathematically satisfying as
well as familiar configurations through experimentation.
[0168] The education pack provides tasks that link and align to
education learning targets according to local and national
requirements.
[0169] It is possible to enhance the `look and feel` of the shape
set further by examining various methods for adding designs to the
two opposite `surfaces` of each and every shape, both the top and
bottom: [0170] (i) Plain [0171] (ii) Symbols--individual symbols on
each shape, as shown for example in FIG. 31a. [0172] (iii)
Image--individual images on each shape, as shown for example in
FIG. 31b. [0173] (iv) Pattern--a pattern that works when two or
more shapes are placed together in any configuration, as shown for
example in FIG. 31c. [0174] (v) Picture puzzle--an image that only
becomes clear when the shapes are placed together in a specific
configuration, as shown for example in FIG. 31d. [0175] (vi)
Application orientated--markings or colours that are essential to
the usage of the shapes in an Application.
[0176] The shape configuration in FIGS. 34a, 34b, 34c and 34d
represents the most elegant embodiment and combination of shape
configuration, shape placement and additional pattern design. The
shape placement template for PS47 is shown in FIG. 34a. The
template comprises of seven bounded areas or sub-regions in each of
which a subset of the shapes may be placed. Conveniently each area
may be provided as a recessed region. The shape configuration
labelled PS47 and demonstrated in FIG. 34b, requires all twenty
three original shapes and omits the original hexagon. Individual
designs are added to each shape in FIG. 34c. A single design
pattern is added to the overall shape configuration in 34d, in
which component parts of the pattern are provided on respective
shapes.
[0177] As each shape has two surfaces, a basic set may exhibit any
two of the above. For example a set might have one overall pattern
that works however the shapes are configured on one side while the
other side exhibits a specific symbol for each individual shape
(iv) & (ii). Another example might use a picture that covers
one side and only works when the shapes are configured in one
unique configuration while the other side might be left blank (v)
& (i). Colour may be added to all variations above as
required.
[0178] The above embodiments represent the realisation of
geometrical apparatus that derive from the footprint of a regular
unit hexagon. This unique and finite set of shapes configure
together to produce various solutions that are both regular and
irregular in form. The properties of the shapes and the way in
which they combine advantageously give rise to the creation of
numerous applications both physical and digital. Many of the
applications can be enjoyed using various media as the delivery
mechanism. All applications and activities that derive from this
invention are `shape` orientated as opposed to `word` or `number`
orientated. Comparison of shapes in order to associate `value or
`hierarchy` is defined either by the internal area of a shape or by
an alternate mathematical property.
* * * * *