ORIGINS OF CATHEDRAL ARCHITECTURE

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The ribbed vault, invented by European Christians, used ribs to support the ceilings above where barrel vaults intersect at right angles in cathedrals. Muslims never used ribbed vaults, but only used the Roman groin vault.
Muslims did not invent the horseshoe arch, but did invent a decorative arch called the scalloped arch.
Another decorative item invented by Muslims was the Lombard band.

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cathedrale_de_reims_-_nef

Muslims did not invent  the ribbed vault or the pointed arch
Muslim contributions to the design of the cathedrals was a few minor decorative items and a small role in the history of the pointed arch. [see the Pointed Arch post]

ROMAN TRANSVERSE ARCHES
The Roman sewer system (connected to the public toilets), construction completed in the 6th century BC, consisted of barrel vaults (also called tunnel vaults) reinforced with transverse arches.
The architecture used in it was invented by the Etruscans of northern Italy.
A photograph can be found at http://www.bsr.ac.uk/la-cloaca-maxima-tra-la-subura-e-il-foro-romano-le-nuove-indagini/cloaca-maxima
Additional photographs of it can be found by doing a Google image search on “Cloaca Maxima” or “Roman sewers”.

CHRISTIAN RIBBED VAULTS
Ribs supported the ceilings above where barrel vaults intersect at right angles in Gothic cathedrals and in many Romanesque cathedrals. This is called the ribbed vault. Muslims never used ribbed vaults, but instead used the Roman groin vault.
A diagram showing a Roman groin vault (2) and a ribbed vault (3) can be seen at http://www.google.com/imgres?imgurl=http%3A%2F%2Fwww.brynmawr.edu%2FAcads%2FCities%2Fwld%2F01410%2F01410b.jpg&imgrefurl=http%3A%2F%2Fwww.solopassion.com%2Fnode%2F948&docid=UdJSR4LwHDicXM&tbnid=22hb5qAMZhxHnM%3A&w=600&h=335&bih=295&biw=534&ved=0ahUKEwik3vmam9TOAhVW12MKHRr4Af8QMwgnKAswCw&iact=mrc&uact=8

THE RIBAT OF SUSA
The Ribat of Susa was built by Muslims in the year 822, in a town that had previously been part of the Byzantine Empire.
In the muslimheritage.com article about it, they showed a photograph of a groin vault with no ribs, and incorrectly labeled it as a ribbed vault.
[Muslims never used the ribbed vault.]
I informed them that it was a groin vault. They then replaced the photograph with a photograph from the Ribat of Susa of a transverse arch reinforcing a barrel vault, similar to those used thirteen centuries earlier in building the ancient Roman sewer system, labeling it as “the early use of the ribbed vaulting“.
It shows a transverse arch, not a ribbed vault. Copying from what Romans had built 1300 years earlier does not make it an “early use” of the transverse arch.

DOMES
Roman domes were often composed of stone arches meeting at the top (often with a with a round hole at the top). The space between the stone arches was concrete made from volcanic ash (called pazzolana), lime, sand and rocks.
Muslims copied the Roman method, but the Muslim did not know how to make concrete, and instead used brick.

THE PALACE OF AL-UKHAIDIR
The Fortress of Al-Ukhaidir (also called the Abbasid palace of Ukhaidir), built in the year 775 in Iraq, did not contain any rib vaults. It contains only a barrel vault (tunnel vault) reinforced with transverse arches, like those used in the .6th century BC Roman sewer system. See the photograph at http://www.gideonbodden.nl/irakfotos/ukhaidher2.jpg

BLIND ARCHES
Blind arches (shallow filled-in arches) were not a Muslim invention, and had been built by the Romans in the 5th century in Ravena Italy, as shown in the first photograph at http://en.wikipedia.org/wiki/Mausoleum_of_Galla_Placidia
of the Mausoleum of Galla Placidia, also known as the oratory of the Church of the Holy Cross.

LOMBARD BAND

See example at https://commons.wikimedia.org/wiki/File:Giulia3.JPG#mw-jump-to-license
This Muslim decorative use of a band of blind crossed arches is the origin of the Lombard band, used in some Romanesque and Gothic cathedrals, but the Christians usually did not cross their arches.

MOORISH ARCH
The horseshoe arch (also called the Moorish arch or the keyhole arch) was used in pre-Islamic Syria in the fourth century.
The horseshoe arch was used by Christians in the 7th century Church of San Juan Bautista, Baños de Cerrato in what is now Syria, and later used by  Visigoths and in India before Muslims used it.

It can be in the shape of more than half of a circle, bulging outward at first before curving inward. It is a decorative innovation, having no functional or structural advantage over ordinary arches.

SCALLOPED ARCH
The scalloped arch was a
decorative innovation of Muslim Spain. Catholics in France reduced the number of scallops to 3 and called it the trefoil arch.

ISLAMIC ARCHITECTURE
Christians did not copy from the style known as “Islamic architecture”. Instead, “Islamic architecture” copied from Christian churches and the Hagia Sophia.

Image by Louis-Kenzo Cahier, via Wikimedia Commons.
image credkit https://commons.m.wikimedia.org/wiki/File:Cath%C3%A9drale_de_Reims_%E2%80%94_Nef.jpg#mw-jump-to-licensehttps://commons.m.wikimedia.org/wiki/File:Cath%C3%A9drale_de_Reims_%E2%80%94_Nef.jpg#mw-jump-to-license

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