in fig1 a to e , different simple forms of construction of the central body are shown , which are bordered by a number of regular congruent triangles , quadrilaterals or pentagons . the known platonic bodies tetrahedrons 1 , cubes 2 , octahedrons 3 , dodecahedrons 4 and icosahedrons 5 result from this . a sphere 26 shows a limit condition in fig1 f . it also is suitable as a central body . the group of , in each case , associated , mutually congruent covering bodies 9 is further examined below . further central bodies result from the bodies 1 to 5 , by cutting back all their corners up to a certain fraction of the length of the edges . in fig2 a , b , c , three examples are shown from the multiplicity of the further possibilities derived from the simple forms of construction of the central body , to which all the bodies known as archimedian solids belong . if , for instance , in a cube all the corners are removed up to a fraction of b = 50 % of the edge length , there arises as in fig2 a a so - called middle crystal 6 . at a fraction of b = 25 %, there appears a corresponding intermediate body 7 as in fig2 b . if all the corners of a icosahedron 5 are removed to 33 %, the truncated icosahedron 8 shown in fig2 c arises . by continuous changing of the fraction b between 0 and 50 % there appear for each of the platonic bodies a multiplicity of unendingly many different central bodies , whose outlines comprise symmetrically arranged , regular polygons with different numbers of corners . it is common to all these central bodies arising in this way , that all of their corners lie on a circumscribed sphere , not drawn here . to these central bodies belong two groups of covering pieces 9 , each mutually congruent , for example for the middle crystal 6 , a first group of eight covering pieces 9 with an equilateral triangle as the basic surface , and a second group of six covering pieces 9 with a quadrilateral of equal length sides as the basic surface , which is also discussed in greater detail below . fig3 shows , in a perspective view , a partly disassembled body according to the invention with a cube 2 as the central body . of the six covering pieces 9 one is here taken out from the body , and is depicted in this fig3 lying alongside the no longer complete body . the opening thus arising provides a clear view onto the surface of the central body , thus here onto one of the squares of the cube 2 , onto the side surfaces of the covering piece 9 removed and onto some of the side surfaces of the remaining covering pieces 9 still joined to the central body . these surfaces can all be printed or otherwise provided with text , pictures , logos or patterns . in fig4 a form of construction of a covering piece 9 is shown , which fits a central body in the form of a dodecahedron 4 . it belongs to an individual group of twelve mutually congruent covering tiles 9 . it is bordered by a basic surface 10 in the form of a regular pentagon , which is essentially congruent with the surfaces of the dodecahedron 4 . this covering piece 9 is further bordered by a regular spherical pentagon as covering surface 11 on the surface of a second sphere , concentric with a circumscribed sphere of the dodecahedron , with a greater but otherwise undefined radius , whereby each of the corners of this spherical pentagon lies on one of the five rays , which start in the centre of the sphere and pass through the corresponding corners of the basic surface 10 . between the basic surface 10 and the spherical pentagon shaped covering surface 11 the covering piece 9 is further bordered by five mutually congruent surface pieces 12 - 16 . to improve visibility , the surface pieces 12 , 13 and 14 are shown as partly transparent and the whole covering piece 9 as a hollow body . the use of hollow covering pieces 9 is similarly , however , obviously in accordance with the invention , indeed for the ability to assemble the body the outer shape is important above all . each of these surface pieces 12 - 16 is bordered by one side of the basic surface 10 , by the two distances between the end points of this side and the corners in each case of the spherical pentagon , and by the smaller great circle arc onto the second sphere through these two corners of the spherical pentagon . not drawn , but similarly in accordance with the invention is if the basic surface 10 is formed not as a flat surface , but as a spherical pentagon through these five corner points on the circumscribed sphere of the dodecahedron . this means in other words that the dodecahedron 4 is replaced by a sphere 26 , retaining the remaining spatial division of the group of the covering pieces 9 . conversely the spherical pentagon of the covering surface 11 can be replaced by a flat pentagon through the same five points , or both the covering surface 11 and also the basic surface 10 can be formed as regular flat pentagons , whereby the covering piece 9 receives the form of a truncated pyramid and the surface pieces 12 - 16 receive the form of equilateral trapeziums . the arrangement according to the invention comprises in this form of construction twelve congruent covering pieces 9 of the type described above and a dodecahedron 4 matching them , as the central body . at least one of the covering pieces 9 includes means , to be able to remove this covering piece 9 as the first to be able to be released from the completely assembled body , here for instance a grommet 25 extending outward from the surface . in fig5 is shown a first form of construction of a connection between the basic surface 10 of the covering piece 9 and any desired surface 17 of the dodecahedron 4 . the surface 17 and the basic surface 10 include joining means , with which a firm connection , but releasable with a small exertion of force , between the dodecahedron 4 and the associated covering piece 9 , can be produced . the fastening means comprise here a press stud 18 , which is placed in the centre of the surface 17 . it is lightly sunk into the surface 17 and is firmly joined to it and thus to the dodecahedron 4 . in the basic surface 10 the associated mating piece 19 of the press stud 18 is similarly firmly fixed to the covering piece 9 . by placing the mating piece 19 onto the press stud 18 and a following light pressure of the pieces against each other the dodecahedron 4 and this covering piece 9 are brought into a relatively firm connection , which can however be released again without the exertion of great force . an exchange of the positions of the press stud 18 and the mating piece 19 is naturally also in accordance with the invention , similarly the use of more than one , for instance three press studs . in a further modification , not shown , instead of the press stud 18 and its mating piece 19 a velcro hook and grommet fabric fastener is used , for example such that its hook part is glued to the basic surface 10 and its grommet part to the surface 17 , or vice versa . in a third , likewise not shown , modification , instead of the press stud 18 and its mating piece 19 , magnets with opposing polarities are applied . magnetic forces are also applied in a further modification , in which magnetic foils are glued to the basic surface 10 , whilst on the surface 17 similarly , magnetic foil or simply a layer of a suitable ferromagnetic material is applied , or vice versa . in a fourth modification instead of the press stud 18 and its mating piece 19 a releasable glue band is applied . the matter set out above applies for every central body represented as examples in fig1 or fig2 . each of these central bodies requires covering pieces 9 matching it in form and quantity . each of these covering pieces 9 has n - fold axes of symmetry . each of these covering pieces 9 can now be subdivided into n further , no longer necessarily symmetrical , but still congruent lesser covering pieces 23 , which then form a new , individual group of covering pieces . in fig6 to 8 this is shown on the example of the covering piece 9 described in fig4 which has a five - fold axis of symmetry , for which thus n = 5 . the transfer to other covering pieces 9 is obvious , for which reason a detailed presentation for other covering pieces 9 can be omitted here . a view is shown looking towards the centre point of the sphere , whereby the covering piece 9 , in schematic representation , is presented roughly as a regular pentagon . in fig6 a first further subdivision is undertaken , in which the regular spherical pentagon of the covering surface 11 is divided into five isosceles congruent spherical triangles 20 . in fig7 a second further subdivision is undertaken , in which the regular spherical pentagon of the covering surface 11 is divided into five spherical kite quadrilaterals 21 . in fig8 a third further subdivision is shown into yet more common spherical quadrilaterals 22 . in an analogue manner , where n = 4 , the covering piece 9 can be divided into four congruent , right - angled isosceles spherical triangles , into four congruent spherical squares or into four more general spherical quadrilaterals . in a similar analogue manner , for n = 3 the covering piece 9 can be divided into three congruent , isosceles spherical triangles , into three congruent spherical kites or into three more general spherical quadrilaterals . naturally each of these sub - parts requires its own fastening means , for instance an individual press stud 18 with corresponding mating pieces 19 on the central body , as is indicated in fig6 - 8 by circles 24 . obviously here also any other suitable fixing means desired can be employed , as has been given in the examples above . a further subdivision of the lesser covering pieces 23 into still smaller units or into non - congruent lesser covering pieces 23 is indeed also according to the invention . such a division is however practically of small importance because of the increasing expenditure for its production and the quickly increasing expenditure of time for the playful assembly of the parts into a complete body . in fig9 a fourth further subdivision of the covering pieces 9 into smaller pieces 23 is shown . on at least two of the subdivided pieces 23 one includes at least one projection 27 and the other at least one opposite and equal cut - out 28 , whereby these lesser pieces 23 are formed as interlocking jigsaw puzzle pieces . obviously more than only one projection and one cut - out 27 , 28 are in accordance with the invention . fig1 shows the development 37 of the surface of a dodecahedron with a fifth further subdivision of the covering pieces 9 into non mutually congruent lesser pieces 23 , namely into lesser pieces 29 and 33 . in this subdivision additionally projections and cut - outs arise on the borders of the covering pieces 9 themselves . the twelve covering pieces 9 are assembled as matching combinations of in each case five selected lesser pieces 29 or 33 . for the overall body a total of thirty lesser pieces 29 and thirty lesser pieces 33 are necessary . each of the individually depicted surfaces of the two lesser pieces 29 , 33 , seen facing away from the central point of the overall body , has the form of an isosceles trapezium with basic angles of 72 °, whereby the base is of equal length to each of the legs . the first lesser piece 29 , drawn in white in fig1 , has a cut - out 31 , 32 in one of its legs and in its base and a projection 30 on its other leg . the second lesser piece 33 , drawn in black in fig1 , has a projection 34 , 36 in each of the legs of the trapezium and a cut - out 35 in the base of the trapezium . the shape of the cut - outs and projections is determined thereby , that the cut - outs and projections 30 , 32 , 35 and 36 are essentially congruent , as are the cut - outs and projections 31 and 34 between each other , and that each of the projections 30 , 34 and 36 and can be pushed into the associated cut - outs 31 , 32 and 35 in each case . using forms of the cut - outs and projections 30 - 32 and 34 - 36 which reduce toward the centre point of the overall body , increases the stability of the overall body , and it can be that the use of the inner body can even be dispensed with . the relative positions of the cut - outs and projections 30 , 36 , 32 and 35 and 31 and 34 respectively can be chosen in each case freely within the length of the touching sides of the trapeziums . with the overall body assembled , the side surfaces of the lesser pieces 29 and 33 lie in planes , which also include in each case the centre point of the overall body . if in the assembly of the lesser pieces 29 , 33 a start is made with five lesser pieces 29 according to the development 37 , the whole body can be put together without difficulty . the use of mirror images of the lesser pieces 29 and 33 as shown is naturally also according to the invention and leads to a mirror image of the overall body . suitable divisions of the covering pieces 9 into lesser pieces are also possible with the other central bodies 1 - 3 , 5 , 6 - 8 and 26 and can be realised in an obvious manner , for which reason their presentation can be dispensed with here . as material for the central bodies 1 - 5 , 26 , 6 - 8 and the covering pieces 9 as well as the lesser covering pieces 23 , many metals and a large number of plastics materials come into question as well as natural materials such as , for instance , wood . the techniques for printing , writing or painting of the parts 1 - 5 , 26 , 6 - 8 , 9 and 23 will not be discussed here , but there are many well known and tried processes for this available to the specialist . at least one of the covering pieces 9 and of the lesser pieces 23 includes means , whereby this piece can be easily be taken out from the completely assembled body , for example , a finger opening in its covering surface 11 , a grip extending out from the covering surface 11 or a grommet 25 , extending out from the covering surface 11 as shown in fig4 . it is similarly in accordance with the invention if only one part out of all the covering parts 9 or 23 can be separated out from the central body , whilst the others remain joined to it firmly , for example also because they are made , together with it from a single piece of material .
the present invention relates to assemblable symmetrical bodies consisting of a regular central body and a multitude of partly congruent covering pieces . fasteners are firmly fixed to the central body , or to the basic surface of each of the covering pieces respectively , providing a relatively firm connection between the regular central body and the covering pieces which can be broken and re - established as often as desired using relatively little force . the covering pieces may be shaped like jigsaw puzzle pieces containing projections and matching cut - outs .