
Italian: 
Metģn cumņ in vore il procediment...
Si scomence cuntune cualsisei disposizion iniziāl (assiome),
par esempli cuntun decagon fat di 5 acuilons
Cumņ metģn in vore il procčs di sudivision par ognidun dai triangui par vź cussģ une sudivision plui fisse
Notģn che i gnūfs triangui a tornin a componi frecis e acuilons
In fin ripuartģn dutis lis planelis a la dimension origjināl
e tornģn a meti in vore il procediment une seconde volte.
Cumņ o vin ancje triangui otūs di sudividi.

English: 
Let us carry out the procedure...
We begin with whatever initial configuration we want (the axiom)
for instance, a decagon made up by five kites
Then we apply the subdivision procedure on each triangle, obtaining a more refined subdivision
Look: the new triangles can piece together again kites and darts
Finally, we blow up all the tiles to the original size
and repeat once more the procedure
Now we have to split also some obtuse triangles

Italian: 
La procedure e rispiete i vincui di adiacence fra i tassei
Ripuartģn di gnūf dutis lis planelis a la dimension origjināl.
Tierce iterazion...
Cuarte...
O cjatģn un decagon centrāl compagn a chel de configurazion iniziāl!
Cuinte iterazion...

English: 
The procedure obeys the adjacency constraints between the tiles
again we blow up the tiles to the original size
Third step...
Fourth step...
A central decagon exactly alike the initial configuration has appeared!
Fifth step...

Italian: 
Seste...
Setime...
Otave...
Si pues lā indenant planelant une aree simpri plui grande.
Ve chi il risultāt dopo nūf iterazions

English: 
Sixth step...
Seventh step...
Eight step...
We can go on, tiling a bigger and bigger part of the plane
This is the result after nine steps

Italian: 
In sostance o vin cjatāt un procediment par tasselā dut il plan rispietant i vincui di adiacence
Par vie de sielte iniziāl (decagon cun 5 acuilons) la planelazion
e varą une simetrie rotazionāl (cun rotazions di 72 grāts)
e ancje des simetriis assiāls (riflessions rispiet a 5 retis che a passin pal centri)
come che si pues capī cjalant chescj moviments.
Lis simetriis a son in dut 10 e a formin il grup diedrāl "D5" (simetriis di un pentagon).

English: 
In conclusion, we have a procedure which tiles all the plane obeying the adjacancy constaints.
Due to the initial configuration (a decagon made up by five kites), the resulting tiling
will have a rotational symmetry (by 72 degrees)
and also axial symmetries (reflections across five axes by the center)
as suggested by some of the motions that you can see
There are precisely ten symmetries, forming the dihedral group "D5" (symmetries of a pentagon)

Italian: 
Chest grup al pues jessi gjenerāt di une rotazion di 72 grāts ...
... e di une riflession fate rispiet a une rete sielzude in maniere oportune che e passe pal centri.
Si pues ancje cambiā un pōc la forme des FRECIS e dai ACUILONS di mūt di otignī une forme di fantasie
cuntun stīl simil a chel dai disens di Escher
Doprģn chestis gnovis figuris par jemplā il plan...
il risultāt dut cās al ą une struture compagne a chź de tasselazion fate cu lis FRECIS e cui ACUILONS.

English: 
The group can be generated by a rotation of 72 degrees
...and a reflection across a suitable axis by the center
We can also modify a little the silhouette of the DART and the KITE, obtaining new tiles
in the spirit of Escher's artworks
Le we employ the new forms in the tessellation...
The result has the same structure as the KITE and DART tiling

Italian: 
Ve chi il risultāt dopo cinc iterazions de tecniche di deflazion/inflazion
Une alternative interessante si ą invezit sielzint in maniere oportune dōs losanghis diferentis
la animazion nus mostre dut cās une relazion une vore strente cul cās des Frecis e dai Acuilons
Lis losanghis a son dividudis in doi triangui isosselis
che a son compagns di chei za cjatāts, fūr che pes proporzions.

English: 
Here is the result after five steps of deflation/inflation
A remarkable option can be found by a suitable choice of two different rhombi
However, there is a close relation with Kite and Dart, as you can see in the animation
The rhombi are split in two isosceles triangles
which are equal to those we have already seen, apart from the proportions

English: 
We can perform a procedure of deflation/inflation starting from a suitable subdivision of the golden triangles
similarly to the subdivision of Dart and Kite
After six steps of the new procedure of deflation/inflation
starting from a suitable initial configuration
we obtain the tiling with rhombi...

Italian: 
Si pues meti in vore un procediment di deflazion/inflazion basantsi suntune oportune division dai triangui auris
simile a chź za doprade pes Frecis e pai Acuilons
Daspņ sīs iterazion dal gnūf procediment di deflazion/inflazion
scomenēant cuntune oportune configurazion iniziāl
o vin cheste tasselazion fate di losanghis...

English: 
Roger Penrose tried to split a large regular pentagon into six smaller pentagons
which side lenght is the inverse square of the golden ratio of the large side
Repeating the subdivision a few times, some holes remain
Some of them have the shape of a regular pentagon, hence they can be included in the subdivision procedure
on the contrary, some other holes have different shapes: rhombus, crown, star
The three different colors of the pentagons are useful to mark adjacency constraints
which induce a non periodic tiling.
In such a way we get a set of six tiles which have the aperiodicity property
Up to now it is not known if there exists a single tile which has the aperiodicity property!

Italian: 
Roger Penrose al ą provāt a dividi un grant pentagon regolār in sīs pentagons plui piēui
che a ąn il lāt compagn al cuadrāt dal rapuart auri rispiet al pentagon grant.
Cu la ripetizion dal procediment si viōt che a restin des busis
cualchidune e ą forme pentagonāl, e duncje e pues jentrā inte procedure di sudivision,
cualchidune altre invezit e ą une forme diferente: di losanghe, di corone e di stele.
I trź colōrs doprāts pai pentagons nus judin a marcā i vincui di adiacence
che nus garantissin di otignī une tasselazion no periodiche.
In cheste maniere si ą un insieme di sīs tassei che a ąn la proprietāt di aperiodicitāt.
Ancjemņ in dģ di vuź no si sa se e esist une singule planele che e ą la proprietāt di aperiodicitāt!
